SEELEY G. 1
1UDD LIBRARY
LAWRENCE
UNIVERSITY
Appleton,
Wisconsin
«__
CURVES
AND THEIR PROPERTIES
A HANDBOOK ON
CURVES
AND THEIR PROPERTIES
ROBERT C. YATES
United States Military Academy
J. W. EDWARDS — ANN ARBOR — 1947
97226
Copyright 1947 by R
octangular C
olar Coordin
NOTATION
=r.
t^r ini
,-
^ lem
a Tangent and the Rad
m Origin to Tangent.
Lithoprinted by E
i
i = /I.
9
f(s ;(f ) = C
1-
«r,p) = C
well Intrinsic Egua
rlll CONTENTS
nephroid
Pedal Curves
Pedal Equations
Radial Curves
Roulettes
Semi-Cubic Parabola
Sketching
Spirals
Strophoid
Trigonometric Functions ....
Trochoids
Witch of Agnesi
PREFACE
lume proposes to supply to student and teacher
,n properties of plane curves. Rather
U ^ Yc 31 r , 'f Lr!-ormation e vhi C h might be found
useful in the classroom and in engine
alphabetical arrangement is
3 aid in the s
Evolutes, Curve Sketching, and
:s readily understandable. If 1
bfi
i
Stropho:
Space Is provided occasionally for the reader to ir
sert notes, proofs, and references of his own and thus
It is with pleasure that the author acknowledges
valuable assistance in the composition of this work.
Mr. H. T. Guard criticized the manuscript and offered
helpful suggestions; Mr. Charles Roth and Mr. William
HISTORY: The Cycloidal curves, including the Astroid,
;,„ r e discovered by Roemer (1674) In his search for the
be st form for gear teeth. Double generation was first
noticed by Daniel Bernoulli in 1725-
1. DESCRIPTION: The
d is a
hypo
y
loid o
f f
ur
Le roll
radius
four
Lmes as
la
ge-
fixed circle
-oiling upon the ins
(See Epicycloids)
ASTROID
EQUATIONS:
x 1 + y 1 - a 1
[:::::::
= (f)(3 cos
= (f)(3 sin
. METRICAL PROPERTIES:
L = 6a
:ion: (Fig. l) Through P d
i the circle of radius ^
BIBLIOGRAPHY
Edwards J.- Calculus , Macmillan (1892) 337-
Salmon 'g • Higher Flare Curves , Dublin (1879) 278.
Wleleitner, H. : Spezielle ebene Kurven , Leipzig (1908).
dl in i , _ i _ i , _. nans, Green
(1895) 339-
Section on Epicycloids , herein.
HISTORY: The Cardioid Is a member of the lamiiy c
cloidal Curves, first studied by Roemer (1674) ir
vestigation for the best form of gear teeth.
1. DESCRIPTION: The Cardioid is an Epicycloid of
cusp: the locus of a point P of a circle rolling
the outside of another of equal size. (Fig. 3a)
Double Generation: (Pig. Jh) . Let
erated by the point P on the rolling ci
Draw ET-, OT'F, and PT' to T. Draw PP t
through T, P, D. Since angle DPT = |, t
has DT as diameter. Now, PD is parallel
arc T'P = arc T'X. Accordingly,
arc TT'X = 2aB = arc TP .
CARDIOID
t-v,c curve may be described as an Epicycloid in
Thus the cui re «uj uc uc=.-i uc 1 ,. ,
ays: by a circle of radius a, or by one of radius £
I ■,.., as shown upon a fixed circle of radius a.
2. EQUATIONS:
(x 2 + y 2 + 2ax) 2 = 4a 2 (x 2 + y 2 ) (Origin at cusp).
r = 2a(l + cos B), r = 2a(l + sin 0) (Origin at ci
9 ( r 2 _ a 2 ) = 8p 2 . (Origin at center of fixed circ
fx = a(2 cos t - cos 2t)
y= a (2slnt-sln2t)> * =*(-»- e 2 " ) .
r 3 =4apS . s=8a-cos(^).
9R
= 64a'
3. METRICAL PROPERTIES:
2 X - (^ )(.a 2 )
ler cardioid.
scial limacon :
3 parallel.
CARDIOID
rotated'
with
he oardioid be pivo
constant angular ve
ted at the cus
locity, a pin,
P and
fixed straight lin
harmonic motio
n. Thu
a(l + cos 0),
«) = -k 2 (r - a),
fe(r- a) =- k 2 (r- a),
il) crossed parallelograms, joined
CARDIOID
= OD = b; AO = BD = CP = a; BP = DC = c
t all times, an;;le F
ingle COX. Any point
BIBLIOGRAPHY
KeoTO and Paires: Mechanism , McGraw Hill (1931)-
L Sketch and -Model
. v. Press, (1941) 182.
CASSINIAN CURV
2. EQUATIONS:
[(x - a) 2 + y 2 ]'[U + a) 2 n
[Fi = C-a,o) F 2 = (a,o)]
3. METRICAL PROPERTIES:
(See Section on Lemniscate)
I
CASSINIAN CURVES
If. GENERAL ITEMS:
--'en formed by a plane paral-
le l to the axis of the torus
its center, of a Rectanrular [yprrbola.
(d) The points P and P' of the linkage shown in
CASSINIAN CURVES
; the coordinates of Q and P be (p,0) and (r,6),^
.re always at right
c 2 - 4a 2 sin 2 8.
;ely. Since 0, D, and Q 1
angles. This
(O'q) 2 = (DQ) 2 - (DO) 2
The attached Peaucellier cell inyerts the point
P under the property
■..! ; ■■.: ; " ■;..,: "■ . . <
Let
Pig
9,
be Fi,
^N. k 2 ?
it pr
off
rv \ FiC
perpen-
liar
to FiF 2
if the circle
any
radius
h B FiX
. Dr
IV cx
ana
perpen-
die
uiar
CY.
CASSINIAN CURVES
!>iY are focal radii (measured from F
BIBLIOGRAPHY
Salmon G. : Higher Plane Curves , Dublin (1879) 44,126.
willson F. N.: Graphics , Graphics Press (1909) 74. ■
Williamson, B. : Calculus , Longmans, Green (1895) 233,533-
Yates, R. C. : Tools , A Mathematical Sketch and Model
Book, L. S. U. Press (1941) 186.
CATENARY
HISTORY: Galileo w£
s the first to inv
stigate tt
noulli
in I69I obtained i
B true form and ga
re some of
its
properties.
1. DESCRIPTION: Th
perfectly flexible
hanging from two s
inextensible chain
rpports not in the
of unifor
n densi
al lin
2. EQUATIONS: If
T cos <f = ka
3 sh(^) = (f)(e a + e a ) ; y 2 =
CATENARY 3
i. METRICAL PROPERTIES:
A = a-s = 2(area triangle PCB) S x = it(ys + ax)
4. GENERAL ITEMS:
(b) Tangents drawn to the curves y = e , y =
(c) The path of B, an involute c
(e) It is a plane section of the surface of least area
(a soap film catenoid) spanning two circular disks,
Pig. 11a. (This is the only minimal surface of revolu-
CATENARY
section of a sail bounded by two
perpendicular to the plane of t
sail 'is normal to the element and proportional to the
square of the velocity, Fig. lib. (See Routh)
Routh, E. J.: An
p. 310.
Vallis! Edinburgh Trans ." XIV, 625
BIBLIOGRAPHY
14th Ed. under "Curves,
Statics , 2nd Ed. (I896) I fl 458,
Dublin (1879) 287.
HISTORY: Causti
ouetelet, Lagrange, and Cayley.
1. A ^caustic
curve 1
s the
envelope of
Light ra
ys,
emitted from
a radia
nt
S, afte
r re-
refracti
on by
a given curv
e f =
The
caustics by
reflect
on
and refract!
on are
ailed
catacaustic
and dia
aus-
3. The instantaneous
;er of motion of S is T. Thus
ape of normals , TQ, _to the ort
is the evolute of the ortho -
3 locus of P Is the pedal of the reflecting curv
a respect to S. Thus the orthotomic is a curve _sln
to the pedal with double its linear dimensions.
' ; ;..■ ■,.■:,'■
con whose pole is the radiant point. With usual x,y ax.
[radius a, radiant point (c,o)]
E(W 2 - a *Kx E +
lowing forms:
(e) Fig- 15 (f)
With the source S at «,
With the source S on the
the incident and reflected
circle, the incident and
rays make angles with
reflected rays makes angle
the normal at T. Thus the
6/2 with the normal at T.
fixed circle 0(a) of
Thus the fixed circle and
radius a/2 has its arc AB
the equal rolling circle
equal to the arc AP of the
have arcs AB and AP equal.
circle through A, P, T of
The point P generates a
radius a/4. The point P of
Cardioid and TPQ is its ta
this latter circle gener-
gent (AP is perpendicular
ates the Nephroid and the
to TP).
reflected ray TPQ is its
tangent (AP is perpendicu-
lar to tp).
These are the bright curves
seen on the surface of cof-
f ee in a cup or upon the table inside of a napkin ring.
7- 2512 Caustics by. Refraction ( Dlacausties ) at a Line L
ST Is Incident, QT refracted, and S is the reflectic
S in L. Produce TQ_to meet the variable circle drawl
through S, Q, and S in P . Let the angles of inciden.
and refraction be 6i and 8 2 and H =
PS -
PS =
. The
SS
tus
of P
is tl
en an
hyper
bo
La wi
th
S, S
ss/n
PQT a
,"1 1
ty
~mLl
"The
-ays
PQT
-bola is
(UlUl
e, the
(Pig
17)
THE CIRCLE
d) If the
point i
-efleoted
rays are
all noi
- 2 = A COS
29 + B
having
1 DESCRIPTION: A circle is a plane continuous curve all
of whose points are equidistant from a fixed coplanar
2. EQUATIONS:
(x - h) 2 + (y - k) E = a 2
x 2 + y 2 + Ax + By + C = C
. METRICAL PROPERTIES:
L = 2na 2 = 4na 2
4. GENERAL ITEMS:
BIBLIOGRAPHY
leal Monthly: 28(1921) 182,187-
Dayley A.- "Memoir on Caustics", Philosophical Trans -
actions ' (1856) ■
Heath, R. S.: Geometrical Optics (1895) 105-
Salmon G. : Higher Plane Curves , Dublin (1879) 98.
r-cle,
produc
circle divides car:.! line : ; constant; i.e., PA ■ PB
= PD-PC (since the arc subtended by / BCD plus that
subtended by L BAP Is the entire circumference, tri-
angles PAD and PBC are similar). To evaluate this
constant, p, draw the line through P and the center
of the circle. Then (P0 - a)(P0 + a) = p = (P0) 2 - a 2 .
The quantity p is called the power of the point P with
respect to the circle. If p <, = , > 0, P lies re-
The locus of all points P which have equal power I
respect to two fixed circles is a line called the
Fig. 'l8(b).
a point called the radical center , a point having
equal power with respect to each of the circles and
equidistant from them.
Thus to construct the radical axis of two circles,
first draw a third arbitrary circle to intersect the
two. Common chords meet on the required axis.
(b) Si militude . Any two coplanar circles have center
0'' similitude: the intersections I and E (collinear
with the centers) of lines joining extremities of
parallel diameters.
The six centers of similitude of three circles lie t
threes on four straight lines.
nine-point circle of a triangle is its orthocenter.
THE CIRCLE
srseotlng circles and to another mem-
La called a train . It Is not to be
Two concentric circles admit a Stelne
angle subtended at the center by each circle of the
train is commensurable with 360°, i.e., equal to
arcs AXB
BYC,
AZC
(A,B,C c
lline
ar)
' Fe .
Studied
y Archi-
medes, s
me of
its
properti
s are
1. jSb +
BYC =
AZC.
2. Its a
ea eq
uals
the area
of th
3. Clrcl
s ins
EE^3 (-ert,usin g Aas
BIBLIOGRAPHY
Daus , P . H . : College G
Johnson, R. A. : Modern
113.
CISSOID
HISTORY: Diodes (between 250-100 BC) utilized the
nary Cissoid (a word from the Greek meaning "ivy")
finding two mean proportionals between given length
progression. This is the cube-root problem since
x 3 =-). Generalizations follow. As early as 1689,
device for the construction of the Cissoid of Diocl
1. DESCRIPTION: Given t
■ves y = fl (x), y = f B (x)
and the fixed point 0. Le
Q and R be the intersect!
of a variable line throug
the given curves. 1
OP = (OR) - (OQ) = QR
rough 0, and the line L
b) distance from 0. The
the locus of P on the variable
Let the two given
perpendicular 1
ordinary Clsso:
secant through such that OP = r = QR.
The generation may be effected by the inte)
of the secant OR and the circle of radius a t!
L at R as this circle rolls upon L. (Fig. 24)
2. EQUATIONS:
(If b = 0: r = 2a-sin e
Cissoid of Diodes) .
(1 + t 2 )
+ (a + b
1 of Diodes: V(rev. about asymp.) = 2u 2 a 3
x(area betw. curve and asymp. ) = —
>) A family of these
Ilssolds may be generated
Dy the Peaucellier cell
r = (^) S ee 0- 2c-cos 8,
b) The Inverse of the family in (a) is,
center of inversion at 0)
y 2 + x 2 (l - 4c 2 ) = 2cx,
in Ellipse, a Parabola , an Hyperbola if c
respectively. (See Conies, 17 ) .
Q (Newton). The
fixed point A
moves along CA
while the other
edge of the
square passes
fixed point on
the line BC per-
The point Q describes a Strophoid (See Strophoid 5e).
(d) Tangent Construction : (See Fig. 26) A has the
at B moves in the direction BQ. Normals to AC and BQ
at A and B respectively meet in H the center of rota-
tion. HP is thus normal to the path of P.
(g) The Cissold as a roulette : One of the curves is
the locus of the vertex of a parabola which rolls upor
an equal fixed one. The common tangent reflects the
( j ) The Stropn.
ire thr
e cei
ter with
re spec
irele.
of Dioc
r plan
k) The
i of 2
e Lord).
:■ of parallel lire
BIBLIOGRAPHY
Hilton, H. : Plane Algebraic Curves, Oxford (1932) 175,
203.
I I . ■ _ r 1 1 , . I in- I
Salmon, G. : Hi gher Plane Curves , Dublin (1879) 182ff.
I , /'r _ I C' '+• Aral tique , Pari:
(1895) II, 115.
L -J. ;-■:■-•. " 'i.i .'.•■.• ■ .■.._. _.:
Co nell t 1 I ( i4u) 77-
CONCHOID
HISTORY: Nloomedes (about 225 BC) utilized the Cc
(from the Greek meaning ,! shell-like" ) in finding
proportionals between two given, lengths (the cube
The Conchoid of Nloomedes is the Conchoid of a Line
= f(8) and
32 CONCHOID
2. EQUATIONS:
General: Let the given curve be
origin. The Conchoid is
r = f(6) + k.
The Conchoid of Mcomedes (for the figure above
solate
ouble p_c
= > k, i
^el.y.
3- METRICAL PROPERTIES:
dlcular to AX at A meet
in the point H, the cen
of OA. Accordingly, HPi
._ (See Pig. 28). The perpen-
the perpendicular to OA at (
ir of rotation of any point
CONCHOID
i of an Angle XOY by the n
the ruler 2k units
apart. Construct BC
parallel to OX such
that OB = k. Draw BA
perpendicular to BC .
Let P move along AB
while the edge of the
ruler passes through
0. The point Q traces
a Conchoid and when
this point falls on BC
the angle is trisected.
(c) The Conchoid of Nic
Mortiz, R. E.: Univ. of Washington Publications, (1923)
[for Conchoids of r = cos(p/q)e].
Hilton, H.: Plane Algebraic Curves , Oxford (1932).
3
i
. DESCRIPTION:
2. EQUATIONS: Given two surfaces f(x,y,z)
Let P i: Ux,yi,zi) be on
P:(x,y,z) a point on the
y - b - k( yi - b
for all values of
5. EXAMPLES: The cone with ve
ing the curve
fx 2 + y 2 - £z . fx 2 + y 2 -
The cone with ve
yVU r =0
2 - lay
The cone with vertex at (
" r(»-p a+ (:
>rigin containing t
2]f]_ + [g(x-l) + Mr-g)] _ 2( z-3) _ 1=0
(x-l) g + (y-g) g + g(x-l)( g -5)+My-g)( Z -3)-3-( Z -;) g .
Jale, Neelley: Analytic G-eometry , Ginn (1938) 284.
HISTORY: The Conies seem to have bee
Menaechmus (a Greek, c .375-325 BC), t
Great. They were apparently conceive
nous problems of t
smpt t
duplicating the cube , and squaring the circle . Instead
of cutting a single fixed cone with a variable plane,
Menaechmus took a fixed intersecting plane and cones of
varying vertex angle, obtaining from those having angles
<= > 90° the Ellipse, Parabola, and Hyperbola respec-
tively. Apollonius is credited with the definition of
the plane locus given first below. The ingenious Pascal
announced his remarkable theorem on inscribed hexagons
in 1639 before the age of 16.
1. DESCRIPTION: A Conic is the li
moves so that the ratio of 1'
b (the focus) divided by
the plane of fi
dist
line (tl
2. SECTIONS OF A CONE: ConE
of angle p cut by a plane
APFD which makes an angle
ting plane at F. The element
through P touches the sphere
at B. Then
Let ACBD be the
Then if PC is pe
to this plane,
right circular cone
(PF)e
constant as P
aries (a, (J constant). The
a conic according to the
ocus and corresponding dii
ersection of the two plan
NOTE: It is
may be had in
evident now that the thre
(A) By fixi
g the cone and varying the
(B) By fixir
tanfandTarbitrary)? 3 "
With either ch
ice, the intersecting cur\
an Ellipse if a < ft ,
a Parabola If a = f) ,
an Hyperbola if « > (b .
types of conic
38 CONICS
3. PARTICULAR TYPE DEMONSTRATIONS:
rmly remarkable that these spheres, inscribed
5 and Its cutting plane, should touch this
le foci of the conic - and that the directrices
bersections of cutting plane and plane of the
Ax*" + 2Bxy 4
; family of lir
I- 2Bm + Cm s )x 2 + 2(D + Em)x + P = 0.
CONICS
he family which cut the curv
family cuts the curve just once. That is, fort
The Hyper
Hi
5 JUSt
The Ellipse Is the conic for which no line of the family
cuts the curve just once. That is, for which:
5. OPTICAL PROPERTY: A simple demonstration of this out-
standing feature of the Corics is given here in the case
of the Ellipse. Similar treatments may be presented for
the Hyperbola and Parabola.
The locus of points P for
which FiP + F 2 P = 2a, a con-
stant, is an Ellipse. Let
drawn at P. Now P is the
only point of the tangent
line for which FiP + F 2 P is
a minimum. For, consider any
FiQ + F 2 Q > FiR + F 2 R = 2a =
Ax 2 + 2Bxy + Cy £ +
2Dx + 2Ey + F =
e point P:(h,k).
ne (whose equation
e form of a tangent
conic):
_jn
+ B(hy + kx) + Cky
^^nJ^
x + h) + E(y + k)
= o (1)
to the curve, meet- Fig. 36
). Their equations are satisfied by (h,k)
thus:
+ B(h yi + kxi) + Ckxi + D(xi + h) + E(yi+k)
+ F
+ B(hy 2 + kx 2 ) + Ckx E + D(x 2 + k) + E(y 2 +k)
+ F
tly, the polar given by (l) contains these
poin
97826
> CONICS
/(p 2 Pi) ( Pi-Pi.) (P £ Q 2
glvenlhTcon
ic°an
iVmrt
necessarily rectangular) and let the conic (Pig.
ce (not
38b)
through^/nl
ariab
le conic
Ax 2 + 2Bxy + Cy
+ 2Dx + 2Ey + F =
in Qi,Q£. The
locu
of Pi
have intercepts ai,a 2 ; b
,b 2 given as the roots
of
which, with P
Q,iQ 2 harmonic
ally
ides
s the
Ax 2 + 2Dx + P =
and Cy 2 + 2Ey + P =
polar of P 2 .
Prom these
= 2 Q E are in ha
rmoni
progres-
_1_ 1 2D
r D .(.|)(i + i).
11 2E
bi + b 2 P
r E=(-|)(^+^)-
Now the polar of P(0,o) is Dx + Ey + F = C
x(J- +i) + y(i +^-) - 2 = 0.
The family of lines through their interse
This affords a simple and classical cons
point P:
Draw arbitrary secants from P and, by the intersectior
of their cross- joins, establish the polar of P. This
46
CONICS
10. P0INTWISE CONSTRI
OTION OP
A CONIC DETERM
NED BY FIVE
GIVE:) POINTS:
Let the five poin
B be num
bered 1,2,3,1',
arbitrary line
' . Draw an
through 1
,
which would me
t the conic
in the require
1 point 3 i .
. [\/___
»',--'
Establish the
Y,Z and the Pa
wo points
cal line.
\</i\
This meets 2 '3
in X and
'*^~)/l \
finally 2,X me
ts the
/XT ~~^
arbitrary line
through 1
/ V,
in 5' . Furthe
- points are
located in the
same way.
Fig. 1+1
11. CONSTRUCTION OF
TO A CONIC GIVE
1 ONLY BY
FIVE POINTS:
In labelling the
onslder 1 and 3
as having
merp
ed so that the
line 1,3' is
2 o / sj
the
tangent. Points
X, Z are
ctete
rmined and the
Pascal line
dra.
n to meet 1' ,3
in Y. The
nlned as in (lo),
CONICS
12. INSCRIBED QUADRILATERALS: The pai
laterals inscribed t
colline
This if
theorem of Pascal.
13. INSCRIBED TRIANGLES
Fur
Pascal hexagon pro-
duces a theorem on
inscribed triangles.
For such triangles,
,'
vertices meet their
opposite sides in
three collinear
14. AEROPLANE DESIGN: The c
d of them. To o
CONICS
16. CONSTRUCTION AND GENERATION: (See also Sket
The following are a few selected from many. Ex
(a) String Methods :
15. DUALITY: The Principle of Duality
of the foregoing.
eal's Theorem (1639)
lizes Into the theorer
(1806):
If a hexag on circumsei
lllnear . (This is apparent
polarizing the Pascal
;on.)
CON1CS
(a) Newton's Method:
Based upo
n the
ide
of t
TO pro
jective pencils, the
Newton. Two angles o
f
constant magnitudes
at A and B A po'rr
line. The point of
r
sides describes a
conic through A and
r
Lrcle or line. The c
17. LINKAGE DESCRIPTIO
N: 1
lie
cted
mechanisms (see TOOLS)
For the >bar linkage
shown, forming a vari-
^
!L
AB = CD = 2a ; AC = BD = 2b
(AD)(BC) =4(a £ - b 2 ).
^Z^^yy^
"@
A point P of CD is
selected and OP = r
drawn parallel to AD
Tig- 52
and BC . OP will remain
parallel to these line
d
int
Let OM = c, MT = z,
wh
re
M is the midpoin
of
Drdins
CONICS
= 2(BT)cos 9 = 2(a - z)cc
l with r = 2(c + z)cos e 1
If now an inversor OEPFP ' be a
Fig. 53 so that
r-p = 2k, where p
An Hyperbola if c < b.
18. RADIUS OF CURVATURE:
For any curve In rectangular
|i I d + y 8 ) 3/a
and N 2 = y 2 (l H
The conic y 2 = 2Ax + Bx , where A is the semi-latus
sctum, is an Ellipse If B < 0, a Parabola if B = 0,
i Hyperbola if B > 0. Here
yy' = A + Bx, yy" + y' 2 = B, and y 3 y" + y 2 y' 2 = By 2 .
ius y 3 y" = By 2 - (A + Bx) 2 = -A 2
19. PROJECTION OF NORMAL LENGTH UPON A FOCAL RADIUS:
Pi(l - e cos 0) = A, (A = serai-latus rectum).
focal radius at K. Draw
the perpendicular at K
to this focal radius
meeting the normal in C
? the Parabola, the angles at P and Q a
to a and FiQ = pi. Thus
PH = pi - pi-cos 6 = A = N-cos a.
20. CENTER OP CURVATURE:
= ^ , from (19),
"rom (18),
BIBLIOGRAPHY
Appleton Century (1937)
imetry , D. 0. Heath (1900) 155-
207.
Hall (1936)
Le Analy
tique, Pari
(1895)-
Lmon, G.: Co
^ o=
Geometr
D. C. H
(1900).
jr, McGraw
Hill (1939)
112.
66.
Tools, A
"(1941) 174,
eath (1923
and Model
CUBIC PARABOLA
HISTORY: Studied particularly by Newton and Leibnitz
(1675) who sought a curve whose sv.bnorr.al is inversely
proportional to its ordinate. Monge used the Parabola
1. DESCRIPTION: The curve is defined by 1
y = Ax 3 + Bx g + Cx + D = A(x - a)(x 2 H
f^l-
2. GENERAL ITEMS:
(a) The Cubic Parabola has max-mln. points only 1
B 2 - 3AC > 0.
railroad engi
CUBIC PARABOLA
(f) It is continuous for all values of x, with no
(g) The Evolute of a £ y = x 3 Is
_9_ 2 .2 128,2 e 9 wl 4 3 e g+5 .
3a 2 (x 2 .
125
y)l- -
(a- + ,«)*
(l) Graphical and Mechanical Solutions :
1. Replace x 3 + hx + k = by the sj
Only one Cubic Parabola
sd be drawn for all
3 of the rational transformation
CUBIC PARABOLA
This may be replacec
(y=x 3 , y+m(x+l)=oj. Since
the solution of each
cubic here requires only
straightedge may be at-
tached to the point (-1,0)
modatlng the quantity m.
CUBIC PARABOLA
BIBLIOGRAPHY
. : Tools , A Mathematical Ske
, R. C. : The Tri
L942).
(1941).
■oblem , The Pranklir
Given the angle AOB = JO.
;hus B itself.
or the equivalent system:
y = 4x 3 , y - 3x - a = 0.
Thus, for trisection of
36, draw the line through
(0,a) parallel to the
fixed line L of slope 3-
This meets the curve
CURVATURE
1. DEFINITION: Curvature is
change of the angle of incl
respect to the arc length.
Precisely,
K =f s .
R = K '
—tfnu^ff^
ntl- y" (or -, 0); at a flex
(or -), at a cusp, R = 0.
2. OSCULATING CIRCLE:
a curve is the circle having
with the curve. That is, the
relations:
-— vf.,»i
(x - a) 2 + (y - p) 2 = r 2
(x - a) + (y - fj)y' =
4* _x
(1 + y' E ) + (y - fi)y" = °
x,y,y',y" belonging to the
curve. These conditions
Fig. 60
give:
r = R, a= x - R-si
if, p - 7 + R-cos <p,
mgle. This is also called the
3.
ge
a s
pr
CURVATURE A
tional algeb
ain at P : ( x ,
r THE ORIGIN
Lgin. Let A
yj. As P app
sculating ci
(Newton): We consider only
be the center of a circle
rcle. Now BP = x is a mean
2y V 2
The Quintic y = x
If the curve be given in polar coordinates, through t
pole and tangent to the polar axis, there is in like
The Cardioid
r = 1 cos B ori = ^ ' c ° 3
26 26
62 CURVATURE
4. CURVATURE IN VARIOUS COORDINATE SYSTEMS:
if = y 2 U +y' 2 )
(See Conies, 18) .
5. CUBVATURE AT A SINGULAR POINT: At a singular point of
F - fxy 2 " f xxfyy
That is, if F < there is an Isolated point , if F = 0,
The slopes y' may be determined (except when y' does not
exist) from the indeterminate form— by the approprlat
CURVATURE
6. CURVATURE FOR VARIOUS CURVES:
CHEWS
EQUATION
E
Hyperbola
^.inse-a*
iS
Catenary
"""""^r*.
Cycloid
b =V Say
x= a(t - slnt)
1 " 2S "olold? 61.
Tractrix
e . o-ln sec tp
c-tan<j
SpIraT 1131,
. . a(e«P - 1,
»-""
Legate
3r Lemniscate)
Ellipse
a - +l ..^.^
•¥
Spir™
^^»»
a n r 2
(n+ l)r n "i (n + l)p
Astroia
x f + y 1 . a 1
J(axy) 1 / 3
£-o^loids
p , a sin bep
a(1.1. a )slnl, 9 , (l-b 2 )-p
7. GENERAL ITEMS:
(a) OsculatlnR circles
CURVATURE
Consider at the origin the
< => 2. (See Evolutes.)
■ the length of the
BIBLIOGRAPHY
HISTORY: Apparently first conceived by Mersenne and
Galileo Galilei in 1599 and studied by Roberval, Des-
cartes, Pascal, Wallis, the Bernoullis and others. It
enters naturally into a variety of situations and is
justly celebrated. (See hb and 4f.)
1. DESCRIPTION: The Cycloid is the path of a point of
circle rolling upon a fixed line (a roulette). The
Prolate at.d Curtate Cycloids' are formed if P is not c
struction, divide the interval OH (= «a) and the
icircle NH into an equal number of parts: 1, 2, 3
. Lay off lPi = HI, 2P 2 = HE, etc., as shown.
2. EQUATIONS:
r = a(l - cos t) = 2a-sin 2 (^
(measured from t
(b) L( one aroh ) = 8a (since R = 0, R M = 4a) (Sir Chris
topher Wren, 1658) .
rotation of P. Thus the tangent at P passes through
N) (Descartes).
(d) R = 4a-cos 6 = 4a-sin(|) = 2 (PH) = 2 ( Normal ) .
mated this result, In 1599 by carefully weighing
pieces of paper cut into the shapes of a cycloidal
4. GENERAL I
(a) Its evolute is an equal
Cycloid. (Huygens 1673- )
CYCLOID
b) Since
s=4a-
OS
§), ff = -
c) A Tau
ochrone
of
le problem
.he detern
inat ion
the type Oi
ib j
tial point
ng was fi
bra ted by Hi
in 1687
id later di
ernoulli
Euler,
an
1 Lagrange.
zertical plane to a
the amplitude. Tt
mass, falling on
heights, will rea
Le of radius ^s. The period of
1 period which is independent of
two balls (particles) of the same
jycloidal arc from different
68
CYCLOID
e evolute (or an involute) of a cycloid
a bob B may be sup-
ported at to de-
J
i
y/
W_ "rairiri
T&fl
1
\%5r- :i:r;:demo
x/7
\SS? resistance) would
^<L
I
U>^
be constant for all
Fig. 65
ount equal
time intervals. Clocks designed upon this
P
rinclple we
re short lived.
(
d) A Brach
sLochrone. First
ath along which a parti-
" YY '
1696, the proble
mination of the p
le moves f
om one point in a plane to another, sub-
ject to a specified
force, in the short-
lowing discussion
J
t
k/f f !
1
is essentially the
1 Iff
Jacques Bernoulli.
Solutions were also
Kg. 66 presented by Leibnit
. 1
1 'Hospital.
For a b
dy falling under r-avitj along any curve
b: y - g. y = gt, y = ~r °r * = v -£ ■
t any Inst
»t
, the velocity
of fall is
uniform density. At any depth y, v = / 2gy" . Let
1 layers of the medium be of infinitesimal
iepth and assume that the velocity of the particle
changes at the surface of each layer. If it is to
>ass from P to Pi to p 2 ... in shortest time, then
iccording to the law of refraction:
Thus the curve of descent, (the limit of the polygon
as h approaches zero and the number of layers incroa
accordingly), is such that (Fig. 67):
an equation that may be iden-
tified as that of a Cy cloid .
(e) The parallel projection
of a. cylindrical helix onto a
plane perpendicular to its
axis is a Cycloid, prolate,
curtate, or ordinary. (Mon-
tucla, 1799; Guillery, 1847.) Fls " 6 ' !
(f ) The Catacaustic of a cycloidal arch for a set of
parallel ra perperdicula to its base is composed of
two Cycloidal arches, (jean Bernoulli 1692.)
(g) The isoptic curve of a Cycloid is a Curtate or
Prolate Cycloid (de La Hire 1704).
(h) Its radial curve is a Circle.
(i) It is frequently found desirable to. design the
face and flank of teeth in rack gears as Cycloids.
(Pig. 68).
DELTOID
HISTORY: Conceived by Euler in 1745 in c
1. DESCRIPTION: The Deltoid is a J-cusped Hypocycloid.
The rolling circle may be either one- third (a = ya) or
tvo- thirds (2a = Jb) as large as the fixed circle.
BIBLIOGRAPHY
: Bibl. Math. (2) vl, p.E
,;.. . , Mi ,.ci a: .: am, McGrai
. : Bibl . Math . (3) v2,p
Hill (19^1) 139 •
iblin (1879) 275-
.le, Leipsic (1912) 77-
For the double generation, consider the right-hand
figure. Here OE = OT = a, AD = AT = -^ , where is
center of the fixed circle and A that of the rollir
circle which carries the tracing point P. Draw TP t
T'E, PD and T'O meeting in F. Draw the circumcircle
F, P, and T 1 with center at A'. This circle is tanj
diameter FT' extended pass
— = j . Thus the radius of this smallest circle is ~
Furthermore, arc TP + arc T'P = arc IT'. Accordingly,
2. EQUATIONS: (where a = yo) .
? cos t+COB 2t)
x 2 +y 2 ) 2 +8bx 3 - 24bxy 2 +
(^
R 2 + 9s 2 = 64b 2 . r 2 = 9b 2 - 8p 2 .
p = b-sin Jtf. z = b( 2 e lt + e" 2lt ).
J. METRICAL PROPERTIES:
L = 16b. <p =, it -| . R = -J 2 = -8p.
A=2ttb 2 = double that of the inscribed circle.
4b = length of tangent (BC) intercepted by the curve.
4. GENERAL ITEMS:
(a) It is the envelope of the Simson line of a fixed
triangle (the line formed by the feet of the perpen-
diculars dropped onto the sides from a variable point
(b) Its evolute is another Deltoid.
(c) Kakeya (l) conjectured that it encloses a region
i straight
V
taking all
io least
Spiral.
to (e,o) is
the
family of
ble7Si-folium, C re
■P.).
on: Since T Is the
ter of rotation of
tangent thus passe
diameter through 1
P, TP is normal to
s through N, the ex
intercepted by the
tangent
BC is b
catacau
stic for
'orthopt
ic curve
t of parallel r
;e). It
) the
nt fi
3 giv
xed at the
en lines (a
3 B, C
meet at right
normals to the curve at B, C, and P all meet
point of the circumcircle .
the tangent BC be held fixed (as a tangent)
and the Deltoid allowed to move, the locus of the
cusps is a Nephroid. (For an elementary geometrical
proof cf this elegant property, see Nat. Math. Mag.,
XIX (1945) P- 530.
2 ][y 2 + (x - c)x] = 4b(x - c)y 2
DELTOID
BIBLIOGRAPHY
n Mathematical Monthly , v29, (1922) 160.
. M. S., v28 (1922) 45-
, Crelle (1865) •
, guar . Jour . Math . (1866).
m. d. Math., v3, p. 166; v4, 7.
din . Math. Soc ■ , v23, 80.
Nouv. Ann. (1870).
d, Eduo. Times Reprint (1866).
; Spezlelle .
, Leipsig (1908)
HISTORY: Leibr
1. DEFINITION:
ENVELOPES
"erential equa-
f(x, y ,p) = 0, p-g;*
X Jf
defines n p's (real or imaginary)
* > 1
for every point (x,y) in the plane.
■/- /
F(x,y,c) = 0,
of the nth degree in c, defines
n c's for each (x,y). Thus at-
tached to each point in the plane
n corresponding slopes. Throughout
Jig. 70
the plane some of these curves
together with their slopes may be r
Mil,
some Imaginary,
some coincident. The locus of those
nts where there
are two or more equal values of p,
thing, two or more equal values of
s the envelope of
the family of its integral curves.
ch of its points
a curve of the family. The equation
of
the envelope
satisfies the differential equation
is usually not a
member of the family.
a. double root of
'roni either of tl
[fjfx.y.p]
■lely) , the envelope 1
J F(x,y,c) =
[F c (x,y,c) =
ENVELOPES
y = px + g(p) .
The method of solution is that of
? +*(5) + (^f)(^)-
a aa tec looua, cuspidal and nodal :
1 (1918). For examples, see Cohen, 1
yielding: | y g - l6x| as the envelope.
yielding the parabola /x + /y = +1
3 of lines, the sum of
Hence , (^H^Qj = , and the general so:
tion is obtained from the first factor: 4^ = 0,
f p = 0, a requirement for an envelope.
J. TECHNIQUE: A family of curves may be given in terms
nected by a certain relation. The following method is
proper- and is particular y adaptable to forms which ar
homogeneous in the parameters. Thus
Their partial differentials are
f a da + f t db = and g a da + eb db =
and thus f a = Xg a , fb = Agb,
The quantities a, b may be eliminated among the equati
to give the envelope. For example:
line of constant length moving
with its ends upon the coordi-
a 2 + b 2 = 1. Their differentials
give (-4) da + ("%)db = and
Multiplying the
m g: I + I - 1
given functions
: E : B
e second by
X, by vlrtt
x = a 3 , y = b 3 ,
„|,«.y».
7] an Astrc
(b) Consider co
stant area
axial ellip
jy U3ing ordinary wax paper.
its plane. Fold P over upon the circle
As P 1 moves upon the circle, the
ENVELOPES 79
an Ellipse if P be inside the circle, an Hyperbola if
outside. (Draw CP ' cutting the crease in Q. Then PQ =
P'Q = u, QC = v. For the Ellipse, u + v = r; for the
Hyperbola u - v = r. The creases are tangents since they
bisect the angles formed by the focal radii.)
For the Parabola, a fixed point P Is folded over to
P' upon a fixed line L(a circle of infinite radius).
P'Q is drawn perpendicular to L and, since PQ = P'Q, the
locus of Q is the Parabola with P as focus, L as direc-
trix, and the crease as a tangent. (The simplicity of
this demonstration should be compared to an analytical
method.) (See Conies 16.)
5. GENERAL ITEMS:
1
te
s on t
ie given c
irve; or
"the e
ivelope of
circles
f fl
ed rad
ius tangen
to
the gi
'en curve;
or as
the e
ivelope of
lines pa
allel to th
e tangent
tn
given
curve and
at a con
tant
distan
ce from the
(d
The f
Lrst posit
ve Pedal
of a
given
curve is tl
3f circles
through
he p
dal pc
ra
ius ve
:tor from
.he pedal
poin
as di
ameter .
(e
The f
Lrst negat
ve Pedal
is t
lope of the
11
agh a poin
th
radlu
3 vector from the p
dal
olnt.
(f
If L,
M, N are
Linear fu
ictio
13 of X
,y, the
CO
elope
Lly L-o 2
h 2M-
is the
M 2 = L-N
,1
ENVELOPES
/elope of a line (or cur^
a curve rolling upon a f
Roulette . For example:
/elope arises in the following
ins problem (Pig. 77): Given the
curve P = 0, the point A, hot
force. Let'y = o he the line
of zero velocity.
time path from A t
the Cycloid normal to P =
generated by a circle rolling
upon y = c. However, let the
family of Cycloids normal to
P = generated by all circles
rolling upon y = c envelope the
curve E = 0. If this envelope
3 F = i
BIBLIOGRAPHY
JS, G. I
: Mess. Math ., II (1872).
Clairaut: Mem. Paris Acad . Sci., (1754).
, . ... !■ :■:■■' ■.:.■■.,:!■"
86-100.
Glaisher, J. V. L. : Mess. Math., XII (1882) 1-14 (exam
Hill, M. J. M.: Proc . Lond. Math . So. XIX (1888) 561-
589, ibid., S 2, XVII (1918) 149.
Kells, L. M. : Differential Equations , McGraw Hill (lj
73ff.
Lagrange: Mem. Berlin Acad . Sci., (1774).
Murray, D. A.: Differential Equations , Longmans, Gree
(1955) 40-49.
PI- and HYPO-CYCLOIDS
ial curves were first conceived by Roemer
'i while studying the best form for gear
and Mersenne had already (1599) <3is-
Lnary Cycloid. The beautiful double genera-
Bernoulli in 1725-
as Caustics . Rectif
jrs find forms of the cycloidal
see Proctor). They also occur
ras given by Newton in his
The Hypocycloid is gen-
erated by a point of a
circle rolling internally
upon a fixed circle.
I
2. DOUBLE GENERATION:
Let the fixed circl
82
- A'F =
EPI- and HYPO-CYCLOIDS
-rying the t
(See Fig. 79.) Draw EI', OT'F, £
intersection of TO and FP and draw t
and D. This circle is tangent to the
angle DPT is a right angle. Now sine
T'E, triangles OET' and OFD are isos
arc TP,
Hypocycloid
may be
generated ir
r difference
= (a - b)cos t
= (a - b)sln t
EPI- and HYPO-CHCLOIDS
,me: (dropping
or (a c ) generate the same curve upon a fixed circle of
radius a. That is, the difference of the radii of fixed
circle and rolling circle gives the radius of a third
circle which will generate the same Hypocycloid.
J. EQUATIONS:
st^e:
: 1 Epicycloid,
= 1 Ordinary Cycloid,
■ 1 Hypocycloid.
I
EPI- and HYPO-CYCLOIDS
l»......
= A 2 B £ |
h--
(^
• the Epicycloid
• the Hypocycloid.
|Bp = a
. METRICAL PROPERTIES:
A (of segment formed by one arch and the
enter)
= ** + D-(^ where* has the valu
s above.
R = AB . 0OE B9 „ ^p wlth the foregol
_g values of
k. (9 may he obtained in terms of t f
•om the given
[See Am. Math. Monthly (1944) p. 587 for an
demonstration of these properties.]
3lementary
5. SPECIAL CASES:
Epicycloids: If b = a...Cardioid
2b = a. . .Nephroid.
Hypocycloids: If 2b = a... Line Segment (
3b = a... Deltoid
4b = a. ..Astroid.
See Trochoids)
EPI- and HYPO-CYCLOIDS 85
6. GENERAL ITEMS:
(a) The Evolute of any Cycloidal Curve is another of
3 fori!
' d 9
AB sin Btp. These evolutes are thus Cycloi
similar to their involutes with linear dimensions
tered by the factor B. Evolutes of Epicycloids ar
smaller, those of Hypocycloids larger, than the c
themselves).
an Epi- or Hypocycloid.
(c) Pedals with respect to the center are the Ros
Curves: r = c-sin(n9). (See Trochoids).
(e) The Epieycl
(f) Tangent Con
3 ( S(
■us center of rotation of .P, TP is
.t at P. The tangent is accordingly
lling circle passing through N, the
ally opposite T, the point of conta
of the circles.
BIBLIOGRAPHY
Edwards, J.: Calculus , Macmillan (1892) 337-
Encyclopaedia Brltanniea . 14 th Ed. , "Curves, Special".
Ohrtmann, C. : Das Problem der Tautochronen .
Proctor, R. A.: The Geometry of Cycloids (I878).
Salmon, G. : Higher Plane Curves , Dublin (1879) 278.
Wleleitner, H.: Spezielle ebene Kurven , Leipsig (1908).
KVOLUTES
reputedly originated with
i studies on light.
Apollonius (about
If (<x,p) is this center.
where R is the radius of
curvature, cp the tangential
angle, and (x,y) a point of
the given curve. The quan-
expres
of a single v
iriable whi
M = -g - R cos cp(d<p/ds) - sin <f(— ),
^=^-R=in 9 (d ? /ds) + cos 9 (f).
EVOLUTES
where d 2 = da 2 + dp 2 .
(h) Generally, ■
y S !T! 4(c) f,
3 EVOLUTES
. EVOLUTES OF SOME CURVES:
(a) The Conies :
The Evolute of
The Ellipse: (~f + (2)* » 1 Is (|) 3 + (|f = 1 ,
The Hyperbola: (*)* - (|)* = 1 l s (^ - (^ = 1 ,
Ha = Kb = a 2 + b 2 .
The Parabola: x 2 = 2ky Is x 2 = ^ (y - k) 3 .
:er of Curvature of
EVOLUTES
<^0
7^J 1
\ /A>^
If the x-axls Is tangent at the origlr
Ho = Limit A = Limit (^) . [See Curva
. GENERAL NOTE: Where there is symmetry in the
urve with respect to a line (except for points
sculation or double flex) there will correspond
n the evolute (approaching the point of 3ymmeti
volute). This is not sufficient, however.
f a curve has a cusp of the first kind, its eve
6. NORMALS TO
A GIVEN Ct
RVE:
Phe E
/olute
of
a c
ntain
ng
the
Fo
ample,
the
=arabola y E
(h,
normals
h,k
y 3
+ 2(1
- h)y
- 2k
rom
where y
epre
sents
the c
rdina
es o
the
ee
of
the nor-
mals at
nd a
tTthe
iffee
t- re
hus,
in ge
era
three
yi +
y 2 + y 3 =
.
If we as
b two
of tt
e thr
e no
mals
e c
oin
3ident,
ble r
this cub
c an
1 its
deriv
ative
3y 2
+ 2(1
) =
0, are
h-l+*=.
of the given Parabola: the envelope of its normals. Thi
evolute divides the plane into two regions from which
one or three normals may be drawn to the Parabola. Froir
points on the evolute, two normals may be established.
An elegant theorem is a consequence
The circle x 2 + y 2 + ax + by + e = n
y 2 = x in points such that
If three of these points are feet of c
to the Parabola, then y 4 = and the c
A theorem involving the Cardioid ca
by inversion.
; the Parabola
I
92 EVOLUTES
7. INTRINSIC EQUATION OP THE EVOLOTE :
Let the given curve be s = f ( <p)
with the points 0' and P ' of
its evolute corresponding to
and P of the given curve.
Then, if a is the arc length
of the evolute:
r= P"o-d<f ° * °"
In terms of the tangential
angle p, (since ? = <f + £ ) ,
BIBLIOGRAPHY
Byerly, W. E . : Differential Calculus , Ginn and Co.
(1879)-
Special."
Edwards J.: Calculus , Macmillan (1892) 268 ff.
Salmon, G. : Higher Plane Curves, Dublin (1879) 82 ff.
Wieleitner, H. : Spezielle ebene Kurven, Leipsig (1908)
169 ff-
EXPONENTIAL CURVES
HISTORY: The number "e" can be traced back to Napier and
the year 1614 where it entered his system of logarithms.
Strangely enough, Napier conceived his Idea of logarithms
before anything was known of exponents. The notion of a
normally distributed variable originated with DeMolvre
o England from Prance, eked out a livelihood by supply-
EXPONENTIAL CURVES
2. GENERAL I
+ k(k
i)(k-a)
continuously
EXPONENTIAL CURVES 95
Ls the maximum possible number of inhabitants -
regulated, for instance, by the food supply. A
sral form devised to fit observations involves
tion f(t) (which may be periodic, for example):
£-f(t).*.(n-x) or
velocity. That i
= (^)(1 - e'***)
(/Ti/- 1 = ( e l£ ) = e"
i (or Decay) 1
>. In an ideal e
ise, pestilence,
'al populations
of individuals,
:curs in controlled
t flies and people.
3 governing law as
. THE PROBABILITY ( OR NORMAL,
1^ = e-* £ / g J (Fig.
87b).
(a) Since y' = -xy and y" = y(x 2 - l), the flex
points are (+1, e" 1 / 2 ). (An inscribed rectangle
one side on the x-axis has area = xy = -y'. The
largest one is given by y" = and thus two corr
are at the flex points.)
EXPONENTIAL CURVES
let I' (n) =
Putting n =
EXPONENTIAL CURVES
BIBLIOGRAPHY
, J. P.: Mathematl
, H . : Mathematical S
specifically:
;ir-g for simplicity:
y- yo -+ (f)U + a)
completely independent of
entering the "slot
separated by nail ob-
The collection
will
stogram
approximating
ber of shot in
the
nal to
the coefficien
binomial expan
sion.
FOLIUM
1IST0RY: First dlscussc
3F DESCARTES
FOLIUM OF DESCARTES
. GENERAL:
(a) Its a
BIBLIOGRAPHY
mica , 14 th Ed. under
2. METRICAL
FUNCTIONS WITH DISCONTINUOUS PROPERTIES
may be useful at various times as counter examples to
the more frequent functions having all the regular
1. FUNCTIONS WITH REMOVABLE DISCONTINUITIES:
FUNCTIONS WITH DISCONTINUOUS PROPERTIES 101
hyperbolas
xy = ± 1 form a
.102 FUNCTIONS WITH DISCONTINUOUS PROPERTIES
3 WITH NON-REMOVABLE DISCONTINUITIES:
imit y = -j Limit y = —
X The left and right limits are
(b) y - sin(^) is not
FUNCTIONS WITH DISCONTINUOUS PROPERTIES 103
(c) y- Limit jl.
« (1 + sin 7i x )t - 1
cut has values +1 or -1 else
Limit y - « Left and right
FUNCTIONS WITH DISCONTINUOUS PROPERTIES
3. OTHER TYPES OF DISCONTINUITIES:
(a) y = x x is undefined for
x = 0, but Limit y = 1.
FUNCTIONS WITH DISCONTINUOUS PROPERTIES 105
(b) y = x 3 is undefined for x = 0, but Limit y = 0.
The function is everywhere discontinuous for x < 0.
06
FUNCTIONS WITH DISCONTINUOUS PROPERTIES
(c) By halving the sides
AC and CB of the
isosceles triangle ABC,
and continuing this
process as shown, the
A
X
"saw tooth" path between
A and B is produced.
with constant length.
Tig. 101
curve of this procession
^nate^'measured ?r
om A,
are of the form
:
K •**■, K-l,
FUNCTIONS WITH DISCONTINUOUS PROPERTIES 107
Wcr-
rig. 105
function y = S D^cosfAx) ,
equilateral triangle is trisected, the middle segment
discarded and an external equilateral triangle built
there) . The limiting curve has finite area, Infinite
The determination of length and area are good
BIBLIOGRAPHY
Edwards, J.: Calculus , Macmillan (1892) 235-
Hardy, G. H. : Pure Mathematics, Macmillan (1933)^62^
Kasner and Newman:
and Schuster (1940) .
Osgood, V. F.: Real Variables, £
Pierpont, J.: Real Variables , G:
shots , Steohert (1938)
GLISSETTES
HISTORY: The idea of Glissettes in Si
SOME EXAMPLES:
(a) The Glissette of the vertex P of a rigid a
whose sides slide upon two fixed points A and
arc of a circle . Furthermore, since P travels
circle, any point Q of AP describes a Limacon .
(See 4).
(c) If a point A of a rod, v
given curve r = f(e), the Gl
Moritz, R. E., U. of Wash
1923, for pictures of man
varieties of this family,
If the curve be given by
p = f(9) referred to the car-
ried point P, then
are parametric equations of
the Glissette traced by P.
For example, the Astrold
3 always through
sin 2<J, y = -. sin 2cp
I rolling
lother determlna
; the problem of 011s-
A simple illustration is the
trammel AB sliding upon two
perpendicular lines. I, the
the fixed circle with center
and radius AB. This point
i if this smaller circle
rolling internally upon
GLISSETTES
3 describes an Ellip
envelope of AB i
6. GENERAL ITEMS:
slides on the x,y axes. Tr.
>( x a + y^ + 3a -) . a « .
=V =a*(x 2 + y 2 ).
iter of an Ellipse
xV = (a 2 - y s )(y 2 - b s ).
) A Parabola slides on a straight line toi
a fixed point of the line. The locus of t
simple cloE
3 difference
circle while one arm passes through a
fixed point F. Hie envelope of the
other arm is a conic with F as focus.
(Hyperbola if F is outside the circle,
Ellipse if inside, Parabola if the
circle is a line.) (See Conies 16.)
HYPERBOLIC FUNCTIONS
: Of disputed origin: either by Mayer or by
Riccati in the 18th century; elaborated upon by Lambert
(who proved the irrationality of n). Further investigated
by Gudermann (1798-I85I), a teacher of Weierstrass. He
complied 7-place tables for logarithms of the hyperbolic
functions in 1832.
1. DESCRIPTION: These functions
^.JlTTl
BIBLIOGRAPHY
■ican Mathematical Monthly : v 52, 384.
ln t, ¥. H. : Roulettes and Glissettes , London (1870).
I . : , •. ■■■ ■ i :■ 1 ' ■■■ 'I ''■ ''■' ■• . : '.' " ' ! ■
, 12,13 (1937-8,
Y
\ \
/ y
\
\
\y \^^
y yl"
\^
/^ '^
N \_
K /
",'. TJ,~
* /
/ \
•;:::::,t:"\
y. tonh x-
L 4 hyperbolic functions
. INTERRELATIONS:
(a) Inverse Relations ;
arc cosh x = ln(x + Vx E - l) , x7
arc tanh x = (|)ln[|^] ; x 2 < 1;
cosh 2 x - slnh 2 x = 1; sechx
csch 2 x = coth 2 x - 1;
Dsh(x + y) = cosh x-c
cosn 2 - ^ 2
HYPERBOLIC FUNCTIONS
(c) Differentials and Integrals:
"■■/.-•
- *±2 oosh iz£ .
Lnh Jx = 4sinh x + 3si
HYPERBOLIC FUNCTIONS
Thus the Hyperbolic functions are attached to the
Rectangular Hyperbola in the same manner that the
trigonometric functions are attached to the circle.
4. ANALYTICAL RELATIONS KITH THE TRIGONOMETRIC FUNCTIONS:
HYPERBOLIC FUNCTIONS
3 REPRESENTATIONS:
!•? 1 , 1-3-5
6. APPLICATIONS:
flexible" 1
hlinl
ing
he Catenary, i
ng from two buj
the
f a
(b) These
functi
3ns
play a dominan
role
in el
, the
engin
nt hyperbolic 1
ponential
form o
\e solutions of
es of
problems
satisfies
the di
ffe
-ential equatior
HYPERBOLIC FUNCTIONS
V = V r -cosh x f& + I r .y|-sinh x /yF,
gives the voltage in terms of voltage and cum
.. ....-eater's (1512-159 1 *) projection
from the center of the sphere onto its tangent cylin-
der with the N-S line as axis,
x = 8, T = gd y,
where (x,y) is the projection of the point on the
sphere whose latitude and longitude are <P and 6, re-
spectively. Along a rhumb line ,
the inclination of a straight course (line)
BIBLIOGRAPHY
Kennelly, A. E. : Applic . of Hjp. Functions to Elec. Engr.
Problems , McGraw-Hill (1912)-
Merriman and Woodward: Higher Mathematics , John Wiley
(I896) 107 ff. . . , .
Slater, J. C: Microwave Transmission, McGraw-Hill (1942)
8 ff.
Ware and Reed: Communication Clrcu:
, John Wiley (1942)
INSTANTANEOUS CENTER OF ROTATION and
THE CONSTRUCTION OF SOME TANGENTS
1. DEFINITION: A rigid body moving in any manner what
neous center of rotation. This
direction of motion of any two
points A, B of the body are known.
Let their respective velocities be
Vi and V 2 . Draw the perpendiculars
to Vi and V £ at A and B. The cen-
HA can move toward A or H (since
the body is rigid) and thus all
points must move parallel to Vi .
Similarly, all points of HB move
If two points of a rig
ie instantaneous center
m of any point P of the
50 INSTANTANEOUS CENTER OF ROTATION
. EXAMPLES:
(a) The Ellip
is an Ellip
se.* AH
and BH are
normals
of A and B
lrid°tbu.
H is the ce
iter of
any
point of th
1 to the
perpendicul
ir PT is
(See
Trochoids,
5e.)
(b) The Con
hold*
- A, the
midpoint of
the c ™
moves along
'th/fix
line and P x
a (ex-
through the
fixed
point 0. Th
point
of PiP 2 pas
through h
direction
PiPe.
Thus the pe
pen-
dioulars OH
and AH
locate H th
of rotation
The
perpendiculars to
INSTANTANEOUS CENTER OF ROTATION
?iH and P 2 H at Pi and P g respectively,
of
(d) The Iso
ptic of a
cu
eve i
angle. If t
-lese tang
the normals
is the cent
body formed
by the c
("ee^Usse
of P. Pc
ex of a t
the
xampl
locu
ngle.
two of whos
e sides t
nil
Normals to
these tar
gen
pass through the cer
s of
the circles
arid nak
stant angle
They meet a
t H, the
of rotatior
lo
of H is ace
ordingly
a circle
ngle. Thus HP is normal
!2 INSTANTANEOUS CENTER OF ROTATION
(e) The point Gllssette of a curve is the loot
a point rigidly attached t
Thus HP is normal to the path of
rig. ii9
rigidly attached to a curve that
rolls upon a fixed curve. The
point of tangency H is the center
of rotation and HP is normal to
the path of P. This is particu-
larly useful in the trochoids of
a circle: the Epi- and Hypo-
cycloids and the ordinary Cycloid.
INTRINSIC EQUATK
INTRODUCTION: The choice of reference
ticular curve may be dictated by its ]
from its properties. Thus, a system o:
coordinates will be selected for cur«
point will be expressed in a polar system with the cen-
tral point as pole. This is well illustrated In situa-
tions involving action under a central force: the path
of the earth about the sun for example. Again, if an
outstanding feature is the distance from a fixed point
upon the tangent to a curve - as in the general problem
of Caustics - a system of pedal c
selected.
The equation
of curves in each of these systems,
are altered by c
ertain transformations. Let a transforit
tion (within a j
system) be such
that the measures of length and angle
, i, , ■ :■.. :..:■. ■■: . '. !■■ ■• ■ . . ! ■ '
of singular poir
ts, etc., will be invariants. If a curv
can be properly
defined in terms of these invariants it
equation would 1
e intrinsic in character and would ex-
press qualities
of the curve which would not change frc
Keown and Faires: Mechanism , McGraw-Hill (1931) Chap.
Niewenglowski, B.: Cours de Geome'trie Analytique , I
(Paris) (1894) 347 ff.
Williamson, B.: Calculus , Longmans, Green (1895) 359-
INTRINSIC EQUATIONS
3 WHEWELL EQUATION: The Whewell equation is that
connecting arc length _s and tangen-
tial angle <p, where <p is measured
from the tangent to the curve at the
initial point of the arc. It will he
as the x-axis or, in polar coordi-
'olloi
-/!"<
initial line. Examples
f = a.cosh(^).
is 2 = [1 4- sinh 2 (^)]dx 2 .
)dx = a-sinh(^), and |s = a -tan y ]
INTRINSIC EQUATIONS
sd directly from the Whewell equation hy ]
1. For example,
'or an involute :
: The inclination <? depends of c
vhic
s i
measured
If this point we
re selected
wher
the
tangent i
a perpendicular to
the original
volve the co-
func
ion
f cp. Thus
for example, the
Cardioid may
give
by
ither 'of
.he equations: s =
k-coag) or
(b) Consid
Here tan y
2. THE CESARO EQUATION: The Cesaro equation relates arc
length and radius of curvature. Such equations are
definitive and follow directly from the Whewell equatic
For example, consider the general family of Cycloidal
The arc length: ds 2 = 8a 2 (l - cos
-8a-cos(|) = -Sa-cos®
_
126 INTRINSIC EQUATIONS
3. INTRINSIC EQUATIONS OF SOME CURVES:
Curve
Whewell Equation
Ceearo Equation
Artroid
e = a.coe 2 9
4s 2 + E 2 = 4a 2
Cardioid
s-a-cosCf)
3 Z + 9B 2 = a 2
Catenary
b = a-tan <p
s 2 + a 2 = aB
Circle
s = a-cp
E . a
Ciesoid
B = a( sec 3 cp - 1)
729(B+a) a = a^s+a) 2 + E 2 ] 3
Cycloid
b . a-sin <p
s 2 + B 2 = a 2
Beltoid
B = y COB 3?
Hypo -cycloids
s. a-sin 0,*
Equiangular
s.a.(e^-l)
m (s + a)«E
Circle ^
b = ^s!
2 a-B.E 2
Nephroid
b = 6o-Bin I
4E 2 + B 2 . 5 6o 2
Iractri*
b - a-m Bee „
a 2 + B 2 . a 2 -e-/a
BIBLIOGRAPHY
ins, London, 263-
INVERSION
HISTORY: Geometrical inversion seems to be due to
Stelner ("the greatest geometer since Apollonius") who
indicated a knowledge of the subject in 1824. He was
closely followed by Quetelet (1825) who gave some ex-
amples. Apparently Independently discovered by Bellavitis
in 1836, by Stubbs and Ingram in 1842-3, and by Lord
Kelvin in 1845. The latter employed the idea with con-
spici
elec
1. DEFINITION: Consider the_circle
mutually Inverse with respect to
(0A)(0A) = k 2 .
- coordinates wit
bangular coordinate
(if this product
Is nega
Inverse and lie
on opposi
Two curves are it
utually
has an inverse b
elonging
128 INVERSION
2. CONSTRUCTION OF INVERSE POINTS:
Fig.
125
For the point A inverse to
then from P the perpendicu-
lar to OA. From similar
right triangles
f ^ or («)(«.*■.
the circle through with
center at A, meeting the
circle of inversion in P, Q.
Circles with centers P and Q
through meet in A. (For
proof, consider the similar
isosceles triangles OAP and
PDA.)
3. PROPERTIES:
(a) As A approaches the
definitely.
distance OA increases in-
(b) Points of the circle
of inversion are invariant.
(c) Circles orthogonal tc
the circle of inversion are
(d) Angles between two ci
rves are preserved in magni-
(a) With center of inversion at a focus, the Con
i family of ovals
(a E + X) + (b 2 + X) =
5. MECHANICAL INVERSORS:
, I The Hart Crossed Parall
122 INVERSION
rhombuses as shown. Its
appearance ended a long
convert circular motion
points 0, P, Q, R taken on
a line parallel to the
bases AD and BC* Draw the
circle through D, A, P,
and Q meeting AB in F. By
unanimously agreed Inso
uble. For the Inverslve
property, draw the clrc
through P with center A
Then, by the secant proi
erty of circles,
(0P)(0Q) = (od)(oc)
(BF)(BA) = (BP)<BD).
Here, the distances BA, BP,
thus BF is constant. Ac-
cordingly, as the mechanist!
is deformed, F is a fixed
point of AB. Again,
= (a-U-Ka+b) = a 2 - b £ .
(0P)(0Q)= (OF)(0A) con-
Moreover,
stant
(P0)(PR) =-(0P)(0Q) =b 2
If directions be assign
by virtue of the foregoing
Thus the Hart Cell of four
Peaucellier arrangement of
eight bars.
ism to describe a circl
center of inversion) as
\£
bar is added to each me Chan
ough the fixed point (the
n in Fig. 130.
to the line of fixed points
6. Since the inverse A of A
leads to the theory of polars
7. The process of inversion forms an expeditious
of solving a variety of problems. For example, t
brated problem of Apollonius (see Circles) is tc
:onfiguration is composed
of two parallel lines and
a circle. The circle tan-
elements is easily ob-
tained by straightedge
and compass. The inverse
circle of inversion 1 ) IT* F1 S- ^ 2
this circle followed by an alteration of its radius I
the length a is the required circle.
.
INVERSION
n is a he
lpful means of generating theorem
cal prope
"If two opposite angles of a
quadrilateral OABC are supple
tary it is cyclic." Let this
figuration be inverted with r
\ spect to 0, sending A, B, C 3
)f C A", B, V and their circumcire
/ into the line AC. Obviously,
7— lies on this line. If B be a
*y/)
moves upon a line. Thus
"The locus of the interse
of circles_on the fixed poin
HISTORY: The Involute
utilized by Huygens ir
of clocks without penc
1. DESCRIPTION: An inv
upon the curve. Or, it
string tautly unwound
BIBLIOGRAPHY
.onen, Leipsig (1906)
Oxford (1941)
Hall (1941)
, Houghton- Mifflin (1929)
2. EQUATIONS:
. METRICAL PROPERTIES:
A = •§- (bounded by OA, OP, AP).
GENERAL ITEMS:
(e)
The limit of
a succession of involutes of any
giv
n curve is a
n Equiangular spiral. (See Spirals
Equ
Langular . )
(f)
In 1891, the
dome of the Royal Observatory at
nstructed in the form of the surfa
of
nerated by an arc of an involute of
oir
le. (Mo. Not
ices Roy. Astr. Soc . , v 51, p. 436
(g)
ial case of the Euler Spirals.
(h)
The roulette
of the center of the attached base
INVOLUTES
(l) Its inverse with respect to
the base circle
spiral tractrix (a curve which i
has constant tangent length).
n polar coordir
(j) It is used frequently in the
design of oam_
(k) Concerning its use in the co
nstruction of j
teeth, consider its generation b
together with its plane along a
line, Fig. 135
of the line on the moving
gency always on the c
internal tangent (the
of action) of the two
circles. Accordingly
velocity ratio is transmitl
ntal law
of gearing is satisfied. 1
dvant c
le older f
orm of cycloidal gear tee
h inc
1. velo
city ratio unaffected by
hang.':
2. cons
rL=rt^Mt-ur
asier
4. more
uniform wear on the teet
Q. Monthly , v 28 (1921) 528.
Byerly, W. E. : Calculus,
Ginn (I889) 133-
, 14th Ed., under "Curves,
Huygens , C: Works, la £
ociete Hollandaise des Scie
(1888) 51 1 *.
Keown and Paires: Mechar
ism, McGraw-Hill (1931) 61,
125.
ISOPTIC CURVES
ISOPTIC CURVE
IISTORY: The origin of the notior
Dbscure. Among contributors to ti-
the names of Chasles on isoptios
trochoids (18^7) anli la Hlre on *
(The Orthoptic of the
Hyperbola is the circle
through the foci of the
corresponding Ellipse and
1. DESCRIPTION: The locu
the Isoptlc of the given
A special case of Orthoptics is the Pedal o:
with respect to a point. (A carpenter's square
one edge through the fixed point while the othi
forms a tangent to the curve).
2. ILLUSTRATION: It is well known that the Ortl
the Parabola is its directrix while those of the Central
Conies are a pair of concentric Circles. These are im-
mediate upon eliminating the parameter m between the
equations in the sets of perpendicular tangents that fol-
low:
tic of
of the rigid body formed by the
constant angle at-R. Thus HR is
normal to the Isoptic generated
If. EXAMPLES:
Given Curve
Isoptic Curve
Epicycloid
Sinusoidal Spiral
Two Circles
Parabola
Curtate or Prolate Cycloid
Epitrochoid
Sinusoidal Spiral
Hyperbola (same focus and directrl?)
ISOPTIC CURVES
Given Curve
Orthoptic Curve
Two Confocal Conic
Concentr
Hypocycloid
itslnee
(a-2b) 2 "
UtoU,,*,,*.
a*
li» : ^-4.co
*»
Sinusoidal Spiral
Sinusoid
al Spiral: r . ..„
3B k (|) where
r-n = a 11 cosrfi
729y E ■
l80x - 16
3(x + y) . x 3
8lyV
+ y 2 ) - 36(x 2 - 2x
f + 5y 2 ) +128 =
x2y£ - Wx= + y 3
l8a 2 xy - 2ya 4 =
x + y +
2a =0
NOTE: The a-Isoptlo of the Parabola y = 4ax is the
Hyperbola tan E a-(a + x) 2 = y 2 - lai and those of the
Ellipse and Hyperbola: (top and bottom signs resp.):
t Isoptics).
(these include t
BIBLIOGRAPHY
Duporcq: L'Inter-m. d. Math . (1896) 291-
Encyclopaedia Britannica : UthEd., "Curves, Special."
Hilton, H.: Plane Algebraic Curves, Oxford (1932) 169-
HISTORY: This curve was devised by P. J. Kiernan in 1945
to establish a family relationship among the Conchoid ,
the Cissoid , and the Strophoid ,
1. DESCRIPTION: The center B of the circle of radius a
moves along the line BA. is a fixed point, _c units
distant from AB. A secant is dravn through and D, the
midpoint of the chord cut from the line DE which is
parallel to AB and b units distant. The locus of Pi and
P 2 , points of intersection of 0D and the circle, is the
Kieroid.
LEMNISCATE OF BERNOULLI
Clssold of t
(FiP){F E P) = a £
(XA)(XB) = a 2 .
Thus, take FiP = XB,
LEMNECATE OF BERNOULLI
3. METRICAL PROPERTIES:
L - 4a(l
V (of r 2 =
' 2-5 2-4-9 2-4-6-13
a 2 oos 26 revolved about
2^(2 - J2).
■ ...) (elliptic
3r jp
4. GENERAL ITEt
(a) I
(b) I
s Pedal of a Rectangular Hyperbola «
Inverse of a Rectangular Hyperbola wit
i center. (The asymptotes of the Hyper-
) It is the Sinusoidal Spiral: v n = a n cos n6 for
.) It is the locus of flex points of a family of
LEMNISCATE OF BERNOULLI
i 30 with the polar
thus easily constructed.
(g) Radius of Curvature P:
(Pig. 141) R =~ . The
projection of R on the radius vector
Thus the perpendic
Liar t
It^J
C, the center of c
'
re.
distance. (See Spirals 2g and Jf . )
LEMNISCATE OF BERNOULLI
a; BC = CP = 00 =
3ince £ns;le 30P = 7; alu;;::,
r 2 = (BP) 2 - (OB) 2 =
2a 2 r 4a 2 sin 2 8,
LEMNISCATE OF BERNOULLI
BIBLIOGRAPHY
1 '. ■■ ,.:'-.,-..;:
Phillips, A. V. : Llnkwork for the Lemnlseate , Arn.
Math. I (1878) 386.
Wieleitner, H. : Spezielle ebene Kurven , Leipsig (IS
Williamson, B.: Plffen Ca ulus . Longmans, G
(I895).
Yates, R. C: Tools , A Mathematical Sketch and Moc
Book , L. S. U. Press, (1941) 172.
LIMACON OF PASCAL
HISTORY: Discovered by Etienne (father of
and discussed by Roosrval in 1650.
1. DESCRTFTION:
ri - nttac led to
circle rolling upon
It is the C onchoid of a
circle where the fixed
point is on the circle.
LIMACON OF PASCAL
1'
. GENERAL ITEMS:
(a
It is the Peda
Cardioid.) (Po
Is, p. 188.)
1 of a circ
nt is on the
e with respect to any
circle, the pedal is
al description, see
(b)
Its Evolute is
the Catacat
stic
f a circle for
any
P light.
(0)
It is the Glis
ariable triangl
3 which slid
eleote
rfeen two fixed
(a)
The locus of a
nt angle whose
r of Limacons (
ly point rig
ides touch
ee G-lissett
idly a
btached to a con
'nd 4)! CleS
(e)
It is the Inve
se of a con
cosB +
, Para
1 respect to a
r°2
us. (The Inl^rT
a-cose + k) =
erslonld?! "
an Ellipse
ola, or Hyper-
> k). (See
(f)
It is a specia
Cartesian
Oval
(g) It is part of
(h) It is the Trise
Folium of Descartes
he Orthopti
ctrix if k
line join!
is 38. (Not
ao aunn wh
3 of a Cardioid.
= a. The angle formed
ig (a,o) with any poin
[x-
st- k-
cos2
1-
4a-c
(
a + y2
- 2a
) 2 = k E (
x 2 + y E
(origin
at
ingular
LIMACON OF PASCAL
(i) Tangent Con
he point A of the bar has
Since T is the cent
A while the point of the
r : gldly attached tc
rolling circle, TP
f the bar itself. The nor-
mals to these directions
neet in H, a point of the
irole. Accordingly, HP is
aormal to the path of P and
Lts perpendicular there is
(k) Double Generation : (See Epicycloids.) It may alsc
be generated by a point attached to a circle rolling
inter nally (centers on the same side of the common
tangent) to a fixed circle half the size of the roll-
LIMACON OF PASCAL
i) m
fHHr
generated by
ortic
'&
o the
par=
C and F fixed
plane. CHJD is
llelogram and P
//v
s a j
ed by a
The
circle
%^\/
under Cardioic
BIBLIOGRAPHY
Edwards, J.: Calculus , Macmillan (I892) 349-
;..:- _.:l... .-. .! L-] i i'r.i - : .' , i: :. '■ I !; f ' ' '' .
V;i ,<■■ ■ ,-,'"'. ;■: I '■ . : . . 1 .■■■.'■,
88.
Yates, R. C. : Tools, A Mathematical Sketch and Model
HISTORY: Studied by Huyge
in 1692 showed that the Nephroid is the cata-
r a cardioid for a luminous cusp. Double genera-
first discovered by Daniel Bernoulli in I725.
1. DESCRIPTION: The Nephroid I
The rolling circle may be one-
halves (3a = 2b) the radius of
a 2-cusped Epicycloid,
alf (a = 2b) or three-
the fixed circle.
Fig. 1U6
For this double generation, let the fixed circle
center and radius OT = OE = a, and the rolling
center A' and radius A'T' = A'F = a/2, the latter
Ing the tracing point P. Draw ET', OT'P, and PI" t
Let D be the intersection of TO and FP and draw t
circle on T, P, and D. This circle is tangent to
fixed circle since angle DPT = n/2 ■ Now since PD
parallel to T'E, triangles OET' and OFD are isosc
TD = Ja.
NEPHROID
Accordingly, if P were attached tc
- the one of radius a/2 or the one
same Nephroid would be generated.
2. EQUATIONS: (a = 2b) .
x = b(3cost
3 = 6b-sin(-t;
? = 4b-sin(|).
= 4b :
- 4a 2 ) 3 = 108a V
= 36b 2 .
(r/2) 5 = a 3 . [sta*(|) + oos*<|)]
: 4b.sin(|).
3. METRICAL PROPERTIES: (a = 2b).
L = 24b. A = 127ib 2 .
4. GENERAL ITEMS:
i the envelop
jther Hophroid.
of a Cayley Sextio (a curve
3 of a diameter of the circle
ion: Since T' (or T) is the
Df rotation of P, the normal is
lerefore PP (or PD) . (Fig. 151 ■ )
BIBLIOGRAPHY
Ueulus . Macmillan (1892) 543 ff.
: A Treatise on the Cycloid (1878) .
: Spezielle ebene Kurven , Leipsig (l<
PARALLEL CURVES
HISTORY: Leibnitz was the first to consider Parallel
Curves in 1692-4, prompted no doubt by the Involute
Huygens (1673).
units distan
t from P me
asured alon
parallel to
the given cu
rve. There
are two
For some
values of 1
, a Parallel
t be unlike
the given
curve in app
otally dis-
similar. Not
Ice the pat
of wheels wi
perpendicu-
lar to their
planes.
. GENERAL ITEMS:
1 normals, they
PARALLEL CURVES
(d) All Involutes of a given
:urve are parallel to each
ither (Fig. 148).
(e
allel
difference i
n length
Of
iies of
E EXA
foil
ViPtES
• Illu
trat 10 n 3
sele
ctedfro
-
(a
Curv
8 th
es pa
rallel to the P
parallel to th
e. (See Salmon'
la are o
tral Con
the 6
(b
Th e
Astro
idx f
y l =a l
9k(x 2
+ J z ) - 18
S K + 8k 3
PARALLEL CURVES
PARALLEL CURVES
3 PARALLEL TO THE ELLIPSE:
PARALLEL CURVES
BIBLIOGRAPHY
is, Green (1879) 337;
Fig. 150
A straight line mechanism is built from two propor -
tional crossed parallelograms OO'EDO and OO'FAO. The
rhombus on OA and OH is completed to B. Since 00' (here
the plane on which the motion takes place) always bi-
sects angle AOH, the point B travels along the line 00'.
(See Tools, p. 96.) Any point P then describes an El-
lipse with semi-axes equal in length to OA + AP and PB.
circle «
along th
e line 00',
the instant
aneous center of 1
tion of
ersection C
of OA produced ar
perpendl
cular to 00
at B. This
point C then lie
circle v
ith center C
and radius
twice OA.
The "kite" CAPG is completed with AP = PG and
CA = CG. Two additional crossed parallelograms APMJA
and PMNRP are attached in order to have PM bisect angle
APG and to insure that PM be always directed toward C.
Thus PM is normal to the path of P and any point such as
Q describes a curve parallel to the Ellipse.
PEDAL CURVES
HISTORY: The idea of positive and negative pedal curves
occurred first to Colin Maclaurin in. I7I8; the name
'Pedal 1 is due to Terquem. The theory of Caustic Curves
includes Pedals in an important role: the orthotomic is
an enlargement of the pedal of the reflecting curve with
respect to the point source of light (Quetelet, 1822).
(See Caustics.) The notion may be enlarged upon to in-
clude loci formed by dropping perpendiculars upon a line
1. DESCRIPTION: The locus Ci, Pig. 151(a), of the foot
of the perpendicular from a fixed point P (the Pedal
Point) upon the tangent to a given curve C is the First
Positive Pedal of C with respect to the fixed point.
The given curve C is the First Negative Pedal of Ci.
Fig. 151 M
lsewhere (see Pedal Equations, 5) t
between the tangent to a given curve ar
or r from the pedal point, Fig. 151(b),
corresponding angle for the Pedal Curve,
to the Pedal is also tangent to the cir
iameter. Accordingly, the envelope of tt
PEDAL CURVES
Conversely, the first negative Pedal Is then the
2. RECTANGULAR EQUATIONS: If the given curve be
f(x,y) = 0, the equation of the Pedal with respect to
the origin is the result of eliminating m between the
and its perpendicular roa th 1 ;in: my +
k is determined so that the line is tangent
For example:
The Pedal of the Parabola y 2 = 2x with re
1_
2:-: 3
3. POLAR EQUATIONS: If (r ,8 ) are
the pole:
ms r 1 + ( a )( £j .
imple, consider the Sinusoidal Spirals
irtl .' Differentiating: n(^) = -n-tan ne
162
PEDAL CURVES
But e=e 0+ f-
f = 6 - nB and thus 8
" (n +°1
Nov r Q =r.sin »
=r.co S n6=a
cos 11 n8
or r -a.oo B < a +
l )A n 8 - a.cos
a+OAf-
1m) ] ■
Thus, dropping sub
scripts, the f]
rst pedal with re
spect
3 me where
nx = T
i+i) '
another. Sinusoidal
kth positive pedal
Spiral. The 11
is thus
table t
TIT
iat folloi
(See al
The
\r ny - = a nk cos
^1 where r
Many of the results given in the
be read directly from this last e
Spirals 3, Pedal Equations 6.)
s can
4. PEDAL EQUATIONS
OF PEDALS: Let the gi
r = f(p) and let Pl
pendicular from the
tangent to the pedal
Pedal Equations):
rigin up
Then (S
per-
\7V—
■Pi = f
p)-Pi.
,x ^\
Thus, replac
pedal equati
ng p an
alogs r
n of th
1 Pi by tt
3 pedal T
eir
Kg. 153
LI
= f(r)
P -1
Thus consider the
f(r) =/(IrO. Hene
2 : ofil
Here f
uation
rcle is
p) =/SF
\r* =y
(^ or
r^ap
"I.
a Cardioid. (See P
Equations of su
fashion.
edal Equations
6.)
are formed in s
milai
5. SOME CURVES AND
PEDAL CURVES
THEIR PEDALS:
163
Given Curve
Pedal Point
First Positive Pedal
Circle
Any Point
limacon
Circle
Cardioid
Parabola
Vertex
Ciesoid
Parabola
Eocus
TmSe Yert a ex
See
Central Conic
EOCUS
^"ctole
Conies,
Central Conic
Center
r 2 . A + B-0OS28
Rectangular Hyperbola
Center
Lemniscate
Equiangular Spiral
Pole
Equiangular Spiral
Cardioid (p*a . f)
Pole (Cusp)
c *;;, s :^
Pole
r 5 . ap 3
rW<§) . a
Pole
Parabola
Smusoidal Spiral
Pole
Sinusoidal Spiral
Astroid: x 1 + y 1 = a 1
Center
2r = ± a-stn28 (Quadri-
Parabola
Poot of Directrix
Bight Strophoid
Parabola
teK H™°J^
Strophoid
Parabola
r::pL.
Tr ^clISn° f
Cissold
Ordinary Focus
Cardioid
Epi- and Hypocycloids
Cento-
Roses
PEDAL CURVES
(Table Continued)
Given Curre
Pedal Point
oeltlve Pedal
Deltoid *
Cusp
Simple Folium
Deltoid
Vertex
Double Folium
Deltoid
Center
Trifolium
Involute of a Circle
Center of Circle
Archimedian Spiral
Origin
tf.rt.M.A
fV = a 1 * 11
Origin
^n.i^) m+n . 0OB m, Bln n e
li/d)"
—
(x2 + y2) n/(n-i)
1/2 a Parabola).
6. MISCELLANEOUS ITEMS:
(a) The 4th negat
(b) The 4th positive pedal of r 8 cos(|)6 = a 9 wit
respect to the pole is a Rectangular Hyperbola .
(c) R'(2r £ - pR) = r 3 where R, R 1 are radii of c
PEDAL CURVES
BIBLIOGRAPHY
Hilton, H. : Plane Alg, Curves , Oxford (1932) 166 f f .
Salmon, G. : Higher Plane Curves , Dublin (1879) 99 f f ■
Wieleitner, H. : Spezielle ebene Kurven, Leipsig (190c
101 etc.
Williamson, B.: Calculus , Longmans, Green (1895) 224
PEDAL EQUATIONS
1. DEFINITION: Certain curves have simple equat
selected fixed point and the perpendicular dlst
upon the variable tangent to the curve. Such re
. PROM RECTANGULAR TO PEDAL EQUATION: If t
the pedal equation may t
and the perpendicular f
(f y )o(y-yo)
+ (f x ) (x-x
r * _ [*0(fJ
o +y (fy)o] 2
[(f x ) 2
+ (f 7 )o 2 J
ere the peda
1 point is tal
3. FROM POLAR TO PEDAL EQUATION:
Among the relations: r = f (9 ) ,
, (For example, see 6.)
origin c
PEDAL EQUATIONS
t = Hff) = P'(|f)/r and thus d6/ds = p/r g
Nov p = r.sin y, and dp = (sin T )dr + r(cos if)cH
or *£ _ /P u ar> ,di|.,
ds _ V W l ds ; -
Thus f s = (^)(f) -ft _
Accordingly, K = f 2 = |^ + f^ = (-) (|E) or
5. PEDAL EQUATIONS OF PEDAL CURVES: Let the pedal equa-
tion of a given curve be r = f(p). If Pl be the perpen-
dicular upon the tangent to the first positive pedal of
i = p(— ) (see Fig. 155).
PEDAL EQUATIONS
r.
Accordingly, <f = if and p 2 = r-pi.
In this last relation, p and Pl play the same roles as do
r and p respectively for the given curve. Thus the pedal
equation of the first positive pedal of r = f(p) is
'f-'^l -
same fashion.
6. EXAMPLES: The Sinusoidal Spirals are \r n = a 11 sin n6 | .
»
Curve
Pedal
r-^.'.ion
"TOPR-^
-a
r 2 6ln28+a e ..O
Reot.Hyperbola
rp ■ a 2
-7a 2
-:
r.einO + a .
Line
p = a
»
-1/2
'"T^T
Parabola
p 2 = ar
2 ^>
+1/2
r-cfMi— e,
Cardioid
(jW <"•
1
Circle
pa=r 2
|
+ 2
'-'— '
—
5?
(See also Spirals, 3 and
PEDAL EQUATIONS 169
3 and' corresponding pedal equations are given:
CURVE
POMT
PEDAL EQUATION
Parabola (IE = ka.)
Vertex
a 2 (r 2 -p 2 ) a =p 2 (r 2 +i ta E )(p 2 +W ! )
Ellipse
Eoous
J^ = T
Ellipse
Center
if - r 2 . a 2 + b 2
Hyperbola
Eoous
^=f +1
Hyperbola
Center
?J! . r 2 = a 2 - b 2
Epi- ana Hypocycloids
p 2 = Ar 2 + B **
Astroid
Center
r 2 + 5P Z = a 2
Equiangular (a) Spiral
' Pole
p = r-sin a
Deltoid
Center
8p 2 + 9r 2 - a 2
Pole
P^=rl +B
r" 1 = a m 9 * (SaooM
1854)
Pole
a = 2:femat's Spiral,
Edwards, J.: Calculus , Macmillan (1892) 161.
Encyclopaedia Brltannica , 14th Ed., under "Cur
Dene Kurven (19c
Williamson, B.: Calculus ,
PURSUIT CURVE
PURSUIT CURVE
k -i/k, .(k+i)/k
in 1732.
1. DESCRIPTION: One particle travels along a specified
, curve while another pursues it,
its motion being always 6.1-
:ed toward the first particle
l related velocities.
3 pursuing pai
Lcle
The special case when k
a(3y - 2a) 2
3. GENERAL ITEMS:
travels on a circle.
until 1921 (F. V. Mc
(b) There i
t where the pursued particl
v and A. S. Hathaway).
rf a triangle begin simultaneously to
chase one another with equal velocities. The path of
each dog is an Equiangular Spiral. (E . Lucas and
H. Brocard, 1877) .
among which £, 7) (coordinates of the pursued particle)
ferential equation of the curve of pursuit.
2. SPECIAL CASE: Let the particle pursued travel from
rest at the x-axis along the line x = a, Fig. I56. The
pursuer starts at the same time from the origin with
velocity k times the former. Then
irves defined by the differential equa-
ire all rectifiable. It is an interesting
;stablish this from the differential
BIBLIOGRAPHY
ds = k-do- or dx + dy = k
3 follows: dx 2 + dy 2 = k E ■ [dy - y'dx + (
= k 2 (a - x) 2 (dy') 2
^V^l ,
Encyclopaedia Brit
Special
Johns Hopkins Un:
Luterbacher, J.:
, (1908) 135.
: Dissertation, Bern (1900).
izette (1930-1) 436.
Math , v 3 (1877) 175, 280.
J
T
RADIAL CURVES
HISTORY: The
1. DEFINITION: Lines are drawn from a selected point
equal and parallel to the radii of curvature of a giv
Radial of the given curve.
. ILLUSTRATIONS:
(a) The radius of curvature
157(a) (see Cycloid) is (R
R = 2(PH) = 4a
Thus, if the fixed point be
of the Cycloid (Fig.
RADIAL CURVES
J. RADIAL CURVES OF THE CONICS:
[Ellipse : b 2 > 0;
Hyperbola: b 2 < 0]
4. GENERAL ITEMS:
RADIAL CURVES
Curve
Radial
Ordinary Catenary
Kampyle of Eudoxus
Catenary of Un.Str.
Straight Line
Tractrix
Kappa Curve
Cycloid
Epicycloid
Roses
Trifolium
Astroid
Quadrifolium
tached to the
plane of a curve which rolls upon a fixed curve (wit
obvious continuity conditions).
BIBLIOGRAPHY
1 4th Ed., "Curves, Special."
Fig. 159
and normal at Oi as axes. Let be originally at Oi and
let T:(xi,yi) be the point of contact. Also let (u,v) b
0; 9 and 91 be the angles of the normals as indicated.
Then
3 in(9 + <Pi) - u-cos(<p +
■cost? + T i) - u-slnfcp
may be expressed In terms of OT, the arc length s. Thes
then are parametric equations of the locus of 0. It Is
not difficult to generalize for any carried point.
Familiar examples of Roulettes of a point are the
Cycloids, the Trochoids, and Involutes.
2. ROULETTES UPON A LINE:
(a) Polar Equation : Consider the Roulette generated
by the point Q attached to the curve r = f(a), re-
ferred to Q as pole (with QOi as initial line), as it
rolls upon the x-axis. Let P be the point of tan-
gency and the point 0i of the curve be originally at
0. The instantaneous center of rotation of Q is P and
mgular equation of the roulette
(here the center of the fixed circle) of the Cycloidal
family:
|b P s = A g (r 2 - a 2 )| where A = a + 2b, and
B = 4b(a + b), as the curve rolls upon the x-axis
(originally a cusp tangent).
ROULETTES
The Cardioid rolls on "top" of the line until t
Lengths of Roulettes and Pedal Curves:
[. Let a point rigidly attached to a closed c
Lng upon a line generate a Roulette through o
ing the fixed tanrer,'.
i under one arch of
sd by a circle of r
3 Ellipse rolls
5 Pedal with reaped
3. THE LOCUS OF THE CENTER OP CURVATURE OP A CURVE,
MEASURED AT THE POINT OP CONTACT, AS THE CURVE ROLLS
UPON A LINE:
Let the rolling curve he given by its Whewell
intrinsic equation: 8 = f(f]
Then, if x,y are coordinates
the center of curvature,
are parametric equations of t
locus. For example, for the
Cycloidal family,
and the locus
1 LINK CARRIED BY A
! ROLLING UPON
the carried line
neighboring point Pi carry
le angle d f. Then if a represents
7 = QT + TQi = sir
frequently easily c
of curvature of rolli
itions of the envelope
For example, consider
circle of radius a. H
t^^cos^-,
ordinary Cycloid .
l82 ROULETTES
6. A CURVE ROLLING UPON AN EQUAL CURVE:
'Oils upo
spondlng points In contact, the
whole configuration is a reflec
(Maclaurin 1720) . Thus the
Roulette of any carried point C
is a curve similar to the pedal
with respect to Oi (the reflec-
tion of 0) with double its
linear dimensions. A simple
illustration is the Cardioid.
(See Caustics.)
7. SOME ROULETTES:
Boiling Cur-re
Fixed Curre
Carried Element
Roulette
Circle
Line
Point of Circle
Cycloid
Parana,
Line
Focus
™^ ( ° rdi "
Ellipse
Line
Focus,
Elliptic Cate-
Hyperbola
Line
Focus
Hyperbolic Cate-
^psr 1
Line
Pole
Tracts
In cirfi: of
Line
Center of Circle
Parabola
Cycloidal
Faulty
line
Center
Ellipse
Line
Any Curve
Point of Line
Involute
Any Curve
, 4 ual Curve
Any Point
CU pTdaf ,11 * r ^
ROULETTES
SOME ROULETTES (Continued):
Eollinf! Curve
Fixed Curve
Carried Element
Eoulette
Parabola
Ectual
Vertex
Ordinary Cissoid
Circle
Circle
Any Point
Cycloidal Family
Parabola
Line
Directrix
Catenary
Circle
Circle
Any Line
Epicycloid
Catenary
Line
Any Line
^Pa^abolT a
curvature. They appear in minimal problems (soap films).
pail
irrangement of
i parallelog]
The
, taken equal i
i smaller side
3 fixed to the plane, Fig. 168(a),
intersect on an Ellipse with A and B as foci. The points
C and D are foci of an equal Ellipse tangent to the
fixed one at P, and the action is that of rolling
-Ellipses. (The crossed parallelogram is used as a "quick
On the other hand, if a long bar 3C be fixed
plane, Fig. 168(b), the short bars (extended) n
Hyperbola with B and C as foci. Upon this Hyper
rolls an equal one with foci A and D, their poi
If the intersection of the shorter bars extended,
PL.-. 169(b), with wheels attached, move along the lir
the Roulette of D (or A) is the Hyperbolic Catenary.
Here A and D are foci of the Hyperbola which touches
Cohn-Vossen: Anscha
Encyclopaedia Brita
nTlif
"Curve
pie, Berlir
3, Special
s, v 1 (18*
(1923).
Action of
Kurven, Le
s, Longman
cal Sketch
(1932) 225.
, 14th Ed.
. C. : Scd
9).
Moritz, R.
E.: U. of Wash
: Curves Formed
London (1874).
. H. : Spezielle
Publ.
J22 the
alculu
... Geometric
Chucks,
Wieleitner
169 ff .
psig (1908)
, Green (1895)
20J ff
Yates, R.
, 238.
hemati
and Model
SEMI-CUBIC PARABOLA
HISTORY: ay 2 = x 3 was the first algebraic curve rectifie
(Nell 1659)- Leibnitz in 1687 proposed the problem of
finding the curve down which a particle may descend unde
the force of gravity, falling equal vertical distances
in equal time intervals with initial velocity different
from zero. Huygens announced the solution as a Semi-Cubi
Parabola with a vertical cusp tangent.
DESCRIPTION: The curve is defined by the equation:
y 2 = Ax 3 + Bx 2 + Cx + D = A(x - a)(x 2 + bx + c) ,
which, from a fancied resemblance to botanical items, is
sometimes called a Calyx and includes forms known as
Tulip, Hyacinth, Convolvulus, Pink, Fucia, Bulbus, etc.,
SEMI-CUBIC PARABOLA
= (x-l)(x-2)(x-3) yi = y 2 = (x-l)(x-
Limit /(x-2)(x-;
^1\] x-1
. GENERAL ITEMS:
Slope at
E and Y-axes different).
- I8x) 3 = [54ax + (-fg)r
BIBLIOGRAPHY
SKETCHING
ALOEBRAIC CURVES: f(x,y) = 0.
1. INTERCEPTS - SYMMETRY - EXTENT ar<
2. ADDITION OF ORDINATES:
is often facilitated by the addltl
For example (see also Fig. I8l):
The general equation of second degree :
Ax 2 + 2Bxy + Cy 2 + 2Dx + 2Ey + F = (l
may be discussed to 'advantage in the same manner.
Cy = - Bx - E + /(B 2 - AC)x 2 + 2(BE - CD)x + E 2 - CF, C ^
ve let Cy = yi + y E ,
SKETCHING
Here y 2 2 - (B 2 - AC )x 2 - 2 (BE - CD)>. - E 2 + CF = 0,
an Ellipse if B 2 - AC < 0, an Hyperbola if B 2 - AC >
a Parabola if B - AC = 0. The construction is effects
CD - BE
C D - BE
= B 2 - AC
inclined at Arc tan(^) 1
5. AUXILIARY AND DIRECTIONAL CURVES:
In the neighborhood of t
origin, ± donates and
given curve follows the
Hyperbola y = - — . As
The quantity e
trols the maxi
(See also Fig. 92.)
SKETCHING IS
4. SLOPES AT THE INTERCEPT POINTS AMD TANGENTS AT THE
ORIGIN: Let the given curve pass through (a,0). A line
through this point and a neighboring point (x,y) has
slope:
quantity i 1 ) approaches m, the slope of the tangent
= c + d (J) + e( i) 2 + fx ,
SKETCHING
flnity". Thus it is as
the curve, generally,
tangent. That is,
r
f(x,y) = and y = mx
= 0, then ^ = a n ^ = 0. But if z = -
3duees to the preccdln;-. Accordingly,
3 (1) has two infinite roots if
x 3 + y 3 - 3xy - 0.
If y * mx + k:
(l+m 3 )x 3 + 3m(mk-lh £
+ 3k( m k-l) x + k 3 = 0.
For an asymptote :
and Jm(mk-l) =0 or k = -1.
OBSERVATIONS: Let P n , Q n be polynomial functions of x,y
of the nth degree, each of which intersects a line in n
points, real or imaginary. Suppose a given polynomial
function can be put into the form:
(y - mx - a).P n _ 1 + V, = ° ( 3>
since its simultaneous solution with the curve results
in an equation of degree (n-l). This family of parallel
lines will thus contain the asymptote. In the case of
the Folium just given:
(y + x)(x 2 - xy + y 2 ) - Jxy = 0,
SKETCHING
I (2y+x)(y
3 y = x for an asymptc
infinity; the line y-mx-k=0in particular cuts
twice . Thus, generally, this latter line is an asymptc
For example:
Thus
the three possible asymptotes of a cubic me
curve again in three finite points upon a 1
the four asymptotes of a quartlc meet the c
eight further points upon a conic; etc.
Thus equations of c
pecifie
irves. For example, a quartlc wit
asymptotes
x = 0, y = 0, y-x = 0, y + x =
meeting the curve again in eight points on the Ellipse
x E + 2y 2 = 1, is:
6. CRITICAL POINTS:
(a) Maximum- minis
Dint (a,b) for which (if y"
le). (See Evolutes.)
7. SINGULAR POINTS: The nature of these points, when
located at the origin, have already been discussed to
Properly defined, such points are those which satisfy
That is, foi
Isolated ( hermit ) j
Dde (double point,
1 9 8 SKETCHING
Thus, at such a point, the slope: ^ = - [•—) has the
Indeterminate form - .
Variations In character are exhibited in the examples
which follow (higher singularities, such as a Double
simpier'ones". 6X1 ° n '
8. POLYNOMIALS: y = P(x) where P(x) is a polynomial
(such curves are called "parabolic"). These have the
following properties:
tinuous for all values of x;
line x = k cuts the curve in but one point
I there are no asymptotes or singularities;
I slope at (a,0) is Limlt[^] as x - a;
) if (x-a) k is a factor of P(x), the point (a,o) if
ordinary if k = 1; max-mln. if k is even; a flex if
k is odd ( i 1).
SKETCHING 201
10. SEMI- POL YNOMINALS: y 2 = P(x) where P(x) is a poly-
nomial (such curves are called "semi-parabolic"). In
sketching semi-parabolic
curves, it may be found ex-
pedient to sketch the curve
taking the square root of
the ordinates Y. Slopes at
the intercepts should be
checked as Indicated in (4).
slope at (2,0) is
11. EXAMPLES:
(a) Semi-Polynomi
-x 2 )
f
y 2 . x(x 2 - 1)
y 2 = x(l
x 2 (
f = x 2 (x - 1)
y 2 = x 2 (
- x 3 )
■f
x 3 (
y 2 = x 3 (x - 1)
y 2 = x 4 (i - x 2 )
yC!
- x 3 )
f
x 5 (
y 2 = (l - x 2 ) 3
y 2 = x(x
- i)(x - a)
y 2
x 2 (
y(a 2 + x 2 )=a 2 x : [y = 0] . x 2 y+y 2 x = a 3 : [x= 0, y=0, x+y.O].
y 3 = x(a 2 -x 2 ) : [x+y=0]. x 3 + y 3 =a 3 : [x + y = 0].
x 3 - a(xy + a 2 )=0 : [x-0]. ( 2a - x)x 2 - y 3 = : [x + y = f ].
x¥-a¥ + lV=0.
(x-y) 2 (x-2y)(x-3f) - 2a( x 3
o)(y - c)x 2 = aV.
a 2 (x+y)(x-2y) . : [fo,
x 2 (x+y)(x-y) E + ax 3 ( x-y) - a 2 y 3 = : [x = ± a, x-y+a= 0,x-y = | ,
x+y+f = 0].
U 2 - y 2 )(y 2 - i+x 2 ) - 6x 3 + 5x 2 y + Jxy 2 - 2y 3 - x 2 + 3xy - 1 = '
:) 5 [Cuep].
: 5 [ Osculin-
(o
Singular Pointa:
a(
-x) 2 =x 3 [OUBp].
2) 2 = x(x-l) 2 [Dout
yj
- 2x 2 y - xy 2 + y 2 .
= 2x 2 y + x 4 y - 2x 4
tedPt].
+ 2x 2 + 2xy - y 2 +
[Ol
Sin]
3X - £
id].
SKETCHING
2. SOME CURVES AND THEIR NAMES:
Alysold (Catenary if a = c): aR =
Boydltch Curves (Lissajou) : fx = £
3ee Osgood's Mechanics for figures).
Bullet Nose Curve : ^s - -^ = 1 .
Cartesian Oval: The locus of pointE
i, r E , to two fixed points satisfy tl
i + m-r 2 = a. The central Conies wll]
fecial cases.
y the theory of Riemann sur
3 AMM, v J>k, p 199) ■
inverse of the Roses; a Cot
Folium : The
30lic Paraboloid, a curve
studies of physical optics
2o4 SKETCHING
SOME CURVES AND THEIR NAMES (Continued):
Kampy le of Eudoxus: a 2 x 4 = b 4 (x 2 + y 2 ) : used by
Eudoxus to solve the cube root problem.
Kappa Curve : y 2 (x 2 + y 2 ) = a x .
Lame Curves : (|)" + (*)° = 1. (See Evolutes) .
Pearls of Sluze : y 11 = k(a - x) s -x a , where the expo-
Pirlform: b 2 y 2 = x 3 (a - x). Pear shaped. See this
Poi i
nt
s Spii
al:
0ll ,,
U of
HipT
Rhod
ne
e (Ro
es)
olds.
Semi
Tr
.dent:
xy 2 = 3b 2 (a - x)
x(y * + b 2 ) - aby
x(y 2 - b 2 ) = aby
xy 2 = m(x 2 + 2bx + b 2 +
b 2 xy 2 = (a - x) 3
c 2 xy 2 = (a- x)(b - x) 2
: Urn, Goblet.
: Pyramid.
d 2 xy 2 = (x-a)(x-b)(x-c) :
Serpentine : A projection of the Horopter
planes taken parallel to its axis.
TrartrijTat a^onstant distance from the I
SKETCHING
SOME CURVES AND THEIR NAMES (Continued):
Trident : xy = ax 3 + bx 2 + ex + d.
Trlsectrlx of Catalan : Identical vith the Tsc
ha
usen Cubl
, an,
l'Hospital's Cubic.
us
Trlsectrlx of
rve resembling
Maclaurin: x(x 2 + y 2 )
the Folium of Descart
= a(y 2 -
Tschirnh
.usen
s Cubi
- S r '° 0S 3
a
pTofMlo
Ider
1 of t
tical with the Witch
he Horopter.
rf A S nesi
Vivian! '
3 Cur\
e: The
spherical curv
f, projections
e, Strophold,
3 x = a. si
th
e Hyperbo
a, Le
and Kappa
Oc
t. (1933)
See.
.M.M.:
28 (1921) 141;
38 (1931
BIBLIOGRAPHY
Echols, W. H.: Calculus , Henry Holt (1908) XV.
Frost, P.: Curve Tracing , Macmillan (I892).
Hilton, H.: Plane Algebraic Curves , Oxford (1932).
I Kurve
WLc-le:
SPIRALS
i of £
HISTORY: The inve
with the ancient Greeks. The famous Equiangular Spiral
was discovered by Descartes, its properties of self-
reproduction by James (Jacob) Bernoulli (1654-1705) who
requested that the curve be engraved upon his tomb with
the phrase "Eadem mutata resurgo" ("I shall arise the
same, though changed").*
. EQUIANGULAR SPIRAL:
(b) C
5 polar
aal).
• R = -dT r
(c) Arc Length : £ = (|f)(^f) = (r-cot a) (-5%
us ai:
a = PT, where _s is measured from
!d) Its pedal
aspect to the
(e) Evolutt
5ole
equal E
PC i
angle PCO = a. 0C i
first and all succe
(g) It is, Fig.
/olute
(a) The curve cuts all radii
of a Loxodrome
holding a fixe
compass), from
(h) Its Catacaustic
source at the pole are Equiangular Spirals .
(i) Lengths of radii drawn at equal angles to each
other form a geometric pro gression .
(j) Roulette : If the spiral be rolled along a line,
the path of the pole, or of the center of curvature
of the point of contact, is a straight line .
SPIRALS
(k) The septa of the Nautili
are Equiangular Spirals. The
curve seems also to appear
in the arrangement of seeds
in the sunflower, the forma
i ■ ■ 1 1 . " .■ 1 1 i .'..•;;: i i ".' i I .
; lei th of an nth involute. Then all first In-
b x = (o + f)de = ce + /f(e)de,
where c represents the distance measured along
value for c for all successive involutes:
-/.'■- b
■co 2 /2! + =e 3 /3: + ...+[/ f(e)de,]
. (See Byerly.) Accordingly,
.,„.ei.e!, ,£,
an Equiangular Spiral .
2. THE SPIRALS: |r = ae n | inclui
following: |n =* l| : | r = ae|
Conan
bu
s
bu
lied
particu-
tract
St
Lll
e
ctan
t. He prob-
■• _iVdc.l
Fig.
center. This suggests the descrip-
rolling without slipping
circle, Fig. 187(a). Here OT = AB = a. Let A
art at A', B at 0. Then AT = arc A' T = r = a6 . Thus
describes the Spiral of Archimedes while A traces
i Involute of the Circle. Note that the center of
tation is T. Thus TA and TB, respectively, are
-rmals to the paths of A and B.
le) Since r = a G and r = ae, this spiral has found
wide use as a cam, Pig. 187(h) to produce uniform
linear motion. The cam is pivoted at the pole and
kept in contact with a spring device, has uniform
(f) It
; Inverse of a Reciprocal Spiral \
(g) "The casings of centrifugal pumps , such as the
German supercharger, follow this spiral to allow a
which increases uniformly in volume with each degr
of rotation of the fan blades to be conducted to t
outlet without creating back-pressure." - P. S. Jo
18th Yearbook, N.C.T.M. (1945) 219.
SPIRALS
(h) The ortho-
graphic projection
of a Conical Helix
on a plane per-
pendicular to its
axis is a Spiral
Equiangular Spiral
(Pig. 188).
teclprocal (Varignon 1704) . ( Son
times called Hyperbolic because of its analogy to the
initial line.
SPIRALS
r all circles (cente
"IT. 6
) The area bounded t
irve and two radii
pole describes a Tractrix.
is a path of a Parti
which variea as the cube of tt
Lemniscate 4h and Spirals 3f .
5 y E = a 2 x) (Fermat I636
e distance. (See
o (because of its
ituus (Cotes, 1722). (Similar
(a) The areas of all circular
SPIRALS
21
spect to the pole
^P
of a Parabolic
(c) Its asymptote
is the initial lin
Limit r- sin 8 =
H*
191
Limit ayC sinB _ Q
(d) The Ionic
y
folute : Together j^* ""* mmmmm ^,
1
I
the Whorl is made
with the curve
emanating from a circle drawn aboul
3. THE SINUSOIDAL SPIRALS: r n = a n co:
r 11 = a n sin n8. (n a rational number).
laurin in 1718.
* = (n + l)r»"i ~ (n + l)p
ilch affords a simple geometrical method of con-
tracting the center of curvature.
(a) it
table 1
in integer.
pedals are again
Sinusoidal Spirals,
(f ) A body acted upon by a central force inversely
proportional to the (2n + 3) power of its distance
moves upon a Sinusoidal Spiral.
g) i
n
Curve
-2
Rectangular Hyperbola
-1
Line
-1/2
Parabola
-1/3
Tschirnhausen Cubic
1/3
Cayley's Sextic
1/2
Cardloid
2
Lemniscate
(In connection with this family see also Pedal Equa -
tions 6 and Pedal Curves 3) ■
(h) Tangent Construction: Since r 11 " 1 r' = - a n sin nf
SPIRALS
1. EULER'S SPIRAL: (Also called Clothoi
of an elastic spring.
5. COTES' SPIRALS:
These are the paths
of a particle sub-
ject to a central
?ce proportional
3 the c
. The 1
eluded in the equa-
1. B
0:
the Equian
gular Spiral;
2. A
1 =
the Recipr
ocal Spiral;
1
a-
1
■ a-c
5-?
.....
in n6 (the
inverse of
Roses).
The figure i
The Glissett
t of the Spiral r
of a Parabola
Spiral: r-sln 28 =
American Mathematic al M:.:it,IiJ y : v 25, pp. 276-282.
Byerly, W. E.: Calculus , Ginn (1889) 133-
Edwards, J.: Calculus , Macmillan (1892) 529, etc.
Encyclopaedia Britannlca : 14th Ed., under "Curves,
Special."
Wieleitner, H. : Spezlelle ebene Kurven , Leipsig (I9O8)
247, etc.
Wlllson, F. N.; Graphics , Graphics Press (1909) 65 f f •
STROPHOID
HISTORY: First conceived b
about 1670.
T Barrov
1. DESCRIPTION: Given the
curve f(x,y) = and the
fixed points and A. Let
K he the intersection
able line through 0. The
locus of the points Pi
and P 2 on OK such that
KPi = KP 2 = KA is the
general Strophoid.
2. SPECIAL CASES: If the c
rve f =
.7 ,*
S — ^~eS
1 circle of fixed radius
2 l8 STROPHOID
asymptote) touching it at R. The line AR through the
fixed point A, distant a units from M, meets the circle
in P. The locus of P is the Right Strophoid. For,
(0V)(VB) = (VP) 2
and thus BP is perpendicular to OP. Accordingly, angle
KPA = angle KAP, and so
KP = KA,
the situation of Fig. 196(a).
. This Strophoid, formed when f = i
identified as a Cissoid of a line and a circle. Thus,
Fig. 197, drav the fixed circle through A with center
0. Let E and D be the intersections of AP extended wi
the line L and the fixed circle. Then in Fig. 197(a):
ED = a-cos 2<f sec 9
and AP = AK = 2a-tan e.sitif = 2a-cot 2cfsin <j .
Thus AP = ED,
STROPHOID
3- EQUATIONS:
Fig. 196(a), 197(a):
Fig. 195(h):
Fig. 197(h):
4. METRICAL PROPERTIES:
A (loop, Fig. 196(a)) = a 2 (l + p.
5. GENERAL ITEMS:
1 It is the Pedal of a Parabola with respect
x(x
a) s
2a
x
x *
a + x)
(c) I
t is a s
pecial
Kierc
id.
graph
ic proje
f
Vivia
ni's Cur
ve.
(e) I
he Carpe
nter's
the
ation of
the Ci
(see Cis
).
with
one edge
passir
g
gh the f
point
B (Fig.
198)
while
moves
along the line
TRACTRIX
HISTORY: Studied by Huygens in 1692 and later by Leibni
Jean Bernoulli, Liouvllle, and Beltrami. AI30 called
Tractory and Equl tangential Curve.
Encyclopaedia B
BIBLIOGRAPHY
mica , 14th Ed., under "Curves,
ionrs de Geometrie Analytlque , F
Fig. 199
1. DESCRIPTION: It is the path of a particle P pulled by
an inextensible string whose end A moves along a line.
The general Tractrlx is produced if A moves along any
specified curve. This is the track of a toy wagon pulled
along by a child; the track of the back wheel of a
bicycle.
e P: (x,y) b
along the x
always towar
-axis. Then, s
d A,
mce S the lng
1 ' y
t-T^
- y 2 |
2. EQUATIONS:
s = a -In se<
. METRICAL PR<
A =
' [/"
y E dy (from pa
she™) ]°.
the circle
(V,
= half t
le volume of th
sphere of
(2*
= area o
the sphere of
radius a) .
(e) Schiele' s Pivot : The solution of the problem of
the proper form of a pivot revolving in a step where
the wear is to be evenly distributed over the face
of the bearing is an arc of the Tractrix. (See Miller
and Lilly.)
f) The Tractrix is utili
See Leslie, Craig.)
g) The mean or Gauss cur
erated by revolving the ci
' the
irface
he arithmetic mean of maximum and minimum curvatur
a point of the surface) is a negative constant
1/a). It is for this reason, together with items
) and (d) Par. 3, that the surface is called the
". It forms a useful model in the stud
Wolfe, Eisenhart, G-raustein.)
of geometry. (Se
) Prom the primary definition (see figure), it is
orthogonal trajectory of a family of circles of
istant radius with centers on a line.
224
TRACTRIX
BIBLIOGRAPHY
Craig: Treatise c
Edwards, J.: Calc
Eisenhart, L. P.
n Prelections,
ulus, Macmillan (1892) 357-
Differential Geometry, Ginn (1909)
Encyclopaedia Bri
(1935).
Leslie: Geometric
Miller and Lilly
Wolfe, H. E. : Nor
tannica: 14th Ed. under "Curves,
Differential Geometry, Macmillan
al Analysis (1821).
Mechanics, D. C. Heath (1915) 285.
r Plane Curves, Dublin (1879) 289.
' . , T r J ( 1 1 ' )
TRIGONOMETRIC FUNCTIONS
: Trigonometry seems to have been developed, vi
certain traces of Indian influence, first by the ArabE
about 800 as an aid to the solution of astronomical pi
lems. Prom them the knowledge probably passed to the
Greeks. Johann MUller (e.1464) wrote the first treatiE
De triangulis omnimodis ; this was followed closely by
other
. DESCRIPTION:
Y
\J
Y
\ 1
\ Y
/I
/
/\
I
\ .
vy
\/
J
rr
/ \
j:ttlr.
/ \
\
WS
- 1/
2. INTERRELATIONS:
(a) Prom the figure: (A + B + C = 71)
TRIGONOMETRIC FUNCTIONS
(b) The Euler form :
(o) A Reduction Formula :
c = 2cos(k-l)x-cosx - cos(k-2)x
c = 2sin(k-l)x-cosx - sin(k-2)x
Thus to convert from a power of the sine or cosi
cos n x =(^~) , expand and replace z k +"z k by 2-c
sln n x = (~r^) n , expand and replace z k - z k by 21-
TRIGONOMETRIC FUNCTIONS
(1 - cos
Sx)
+ 3)
2 (1
. 3 (3ein x
- sin 3x)
(c
os3x +
. 4 (cob kx
s ( Bin 5 x-
5sin Jx+lC
l+oos 2x + 3)
(c
os 5x+5c
8
ob 3x+10cos x)
16
16
(e)
2
sin kx
n + 1
n 2 x ■ s
mf
n + 1
111 T
sin -
(f) From the Euler form given In (b) :
3. SERIES:
3 15 315 2835 '
" 3 " i+5 " 9^5 " V725 + "" :
TRIGONOMETRIC FUNCTIONS
, j. ■£. 5x* , 61 T e 277 a
360 15120
. £ *. L± . JL. + 1- 5'5 . *L .
+ 5? " 5? 7x 7 " •••'
arc CBCX = I + I . 1 + ill . _1_ + ill5 . J_ + .^ x2>
. DIFFERENTIALS AMD INTEGRALS:
/«*« — * l-«1
/ — --|o»x -cot, | = m|t m f|.
TRIGONOMETRIC FUNCTIONS
5- GENERAL ITEMS:
(a) Periodicity : All trigonometric func"
periodic. For example:
y = A'sin Bx has period: 4r and ami
leflned by the differential
Its solution is y = A-eos (Bt +9), in which the
arbitrary constants are
A: the amplitude of the vibration ,
9 : the phase-lap: .
(c) The Sine (or Cosine) curve is the orthogonal pro -
jection of a cylindrical Helix , Fig. 203(a), (a curve
cutting all elements of the cylinder at the same
angle) onto a plane parallel to the axis of the
cylinder (See Cycloid 5e.)
Fig 203(b). Let the intersecting plar
TRIGONOMETRIC FUNCTIONS
1 cylinder: (z-l) £ + x 2 = 1
■oils
A worthwhile model of this may be fashioned from s
roll of paper. When slicing through the roll, do r
flatten it.
airplane travels on a
great circle around the
earth, the plane of the
arbitrary cylinder cir-
cumscribing the earth
in an Ellipse . If the
cylinder be cut and laid
flat as in (d) above,
leriod of
9 Theory : Trigo-
This is exhibited 3
Fig. 205.
TRIGONOMETRIC FUNCTIONS
f Prentice-Hall.)
TRIGONOMETRIC FUNCTIONS
Hurler Development of a given function is the
jsition of fundamental Sine waves of ir.croasir
lency to form successive approximations to the
Lbration. For example, the "step" function
BIBLIOGRAPHY
TROCHOIDS
HISTORY: Special Trochoids were first
in 1525 and by Roemer in 1674, the lat
with his study of the best form for ge
onceived by Diirer
r teeth.
1. DESCRIPTION: Trochoids are Roulette
- the locus of a
1 curve that roll
upon
'ixed curve. The r
rer sally applied
Epi-
md Hypotrochoids
jath of a point r
gidly
ittached to a cir
le
-oiling upon a fi>
ed
2. EQUATIONS:
;os(mt/b) x = n-cos t + k-cos(nt/b)
iln(at/b) y = n-sin t - .k- S in(nt/b)
- and Hypocycloids if k = b) .
3. GENERAL ITEMS:
i line (Pig. 208):
(c) The Ellipse is the Hypotrochoid where a = 2b .
Consider generation by the point P [Pig. 209(a)] .
Draw OP to X. Then, since arc TP equals arc TX, P was
originally at X and P thus lies always on the line OX.
Likewise, the diametrically opposite point Q lies al-
ways on 0Y, the line perpendicular to OX. Every point
of the rolling circle accordingly describes a diameter
of the fixed circle. The action here then is equiva-
lent to that of a rod sliding with its ends upon two
perpendicular lines - that is, a Trammel of Archi-
medes. Anjr point F of the rod describes an Ellipse
whose axes are OX and OY . Furthermore, any point G,
rigidly connected with the rolling circle, describes
an Ellipse with the lines traced by the extremities
of the diameter through G as axes (Nasir, about I250) .
the diameter PQ envelopes an Astroid
TROCHOIDS
209(b)
e Double
and tr
Generation
If the small
RX pas
smalle
ses alwa
ys through
. Consider
a
diamet
er. Sine
e SO is a
passes
through a
is a L
i f
rollin
I circle
"described
-Hi
Envel °Pe Roulette: Any line rigidly attached to
the rolling circle envelopes a Circle . (See Llmacon
3k; Roulettes h; Glissettes 5.)
(e) The Rose Curves: r = a cos ne
ircte
r - a sin nfl
are Hypotrochoids crenerateri hy » ,
of radius
2 ( n + !) rolling within a fixed 01
rele
f radius
; units distant from its center. (First noticec
'di in 1752 and then by Ridolphi in 1844. See
aa -tip, P = 2(a
Thus in polar co<
e) =2(a . b) oo S — ^-e.
I (f<
'engelly: Theoretical Naval Arc
study of ocean waves).
Edwards, J.: Calculus , Macmlllan (1892) 343 tt .
Lorla, G. : Spezlelle algebralsche und Transzendente
ebene Kurven , Lelpsig (1902) II 109 .
Salmon, G. : Higher Plane Curves , Dublin (1879) VII.
Williamson, B.: Calculus , Longmans, Green (1895) 3^8 f
philosopher, and somnambulist), appointed profes
Mathematics at Bologna by Pope Benedict XIV. Tre
earlier (before 1666) by Fermat and in 1703 by G
Also called the Versiera.
, VI (1939) 211; VIII (19U) 135 a:
XLTI U9k6) 57.1
1. DESCRIPTION: A sec
on the fixed circle c
ant OA through a selecte
uts the circle in Q. QP
The P path°of la p is the"
diameter OK, AP paralle
WITCH OF AGNESI
3. METRICAL PROPERTIES:
(a) Area between the
times the area of the
(b) Centroid of this
(c) V x = te s a 3 .
(d) Flex points occur
: (0,f).
. .;•;■..■!. <,i. P I 1 .... '• ■ i-
-oduced by doubling the ordinate s of the Witch
irve was studied by J. Gregory in I658 and use
3ibnltz in 1674 in deriving the famous express
Edwards, J.: Calculus , Macmillan (1892) 355-
Encyclopaedia Britannica : 14th Ed., under "Cu
noulli: 1,1
1,93,145,152,175,206,22
ant: 108,175
0,149,151,152,155,
4,223,233,255
2,143,161,165,185,218,2
npass Construction: 128
ichold: 51-53:50,108,109,120,
k: 3>+-3 ;37, 38,39
files: 36-55:20,78,79,87,88,
L12,130, 131, 138,11*0, H9,156,
163, 173,1°?, 189, 195, 203
capital's: 203,205
tola: 56-59:89,186,197
onal Curves: 190
ilnant: 59,57,76,189
Double generation: 81
Duality: 1*8
Durer: 175,233
"e": 93,9**
Elastic spring: 215
Ellipse: 36-55:2,19,27,63,78,79,
88 , 109 , 111 , 112 , 120 , 139 , 11*0 ,
11*9, 157, 158,161*, 169, 173, 178,
179, 180, 182, 183,181+, 189, 195,
202, 299, 230,231*
Elliptic Catenary: 179,182,181*
i tare lopes: 75-80;2,3,15, 50,72,
'('3,85,87,91,108,109,110,111,
112, 135, 139, l'* 1 *, 153, 155,160,
161, 175, 180, 181, 23 1 *, 235
180. 161, 181*, 197, 207, 213, 215,
Epi: 203
Epicycloid: 81-85; '
126,139,152,163,1
ter of: 5 1 *, 55,11*5,150,213
180,182,183
Cusp: 20,27,90,192,197,199,200,
Epitrochoids: (see
Equation of second
Cylinder: 229,230
188
Cycloid: 65-70;l,l*,65, 80,89, 92,
Equiangular Spiral
122,125,126,136,137,138,139,
Equiangular)
172,17l*,176,177,179,l80,l8l,
Equitangential Curv
182,183, (see also Epicycloids
Eudoxus, Eippopede
Kampyle of
da Vinci, Leonardo: 170
Euler: 67,71,82
Deltoid: 71-7'*; Bk ,126 ,lk0 ,16k ,
Euler form: 9>*,ll6,
Lutes: 86-92:2,5,15,16,19,20,
Hathaway: 171
7,66,68,72,79,85,135,139,11*9,
Helix: 69,203,20
?2,155, 155,173, 187, 197,201+,
Helmet: 201*
mential Curros: 93-97;20
Hessian 99
oat: 237
Hippias, Quadrat
rix of: 201*
toiler 1 : (eeespiraiB '
Hippopede of Eud
Hire: 138,175
oxus: 203
c point: 10,56,87,90,196,198
.ubic: 203,205
Huygene: 15,66,6
7,86,135,152,
t. of Descartes: 9 8-99;193,
Hyacinth: 186
um: 72; (Simple, Double,
Hyperbola: 56-55
19,27,63,78,
s: 1,69,81,137,233
no's Lemniscate: 203
8,139,11*9,216
79,88,101,112,115,116,129,130,
159 , 11+0, 11+1+, 11+9, 157, 163, 161*,
168, 169, 173, 182, 181*, 189, 195,
85,87,125,126,155,156,161+,
176,182,183,208,209,222
Isolated point: 192,197,200,
Kakeya: 72
Mercator: 118,230
Kappa Curve: 17! + ,?0'+,205/222
Minimal Surfaces: 13,183
Monge: 56
Kierold: l)+l-ll+2;29,33,219
Kite: 1 5 8
Morley: 171
Lagrange: 15,67,75
Motion, line: 81+, 132, 158,210,
Lambert: 113
23>+
Lame' Curve: 87,l6l+,20l+
Law of Orowth ( or Decay) : 91+
Multiple point : 20,192,197,199,
Leibnitz: 56,68,155,175,186,
Mapler: 93
221,238
Hapkin ring: 17
Masir: 23I+
1)+7;9,10,63, 150, 157, 163, 168,
Neil: 186
nephroid: 152-15l+;17,73,8l+,87,
Light rays: 15,86
Limacon of Pascal: 1U8-151;5,7,
Hewton: 28,51,56,60,67,68,81,
16,31,108,110,121,130,139,11+0,
175
163,231+, 235
Nicomedes, Chonchoid of: 31-33;
Line motion: 8>+, 132,158,210,23)+
108,11+2
Linkages: 6,9,25,51,152,1 K6, 151,
node: 192,197,199,200
158,183
normal Curve: 95,96
Liouvllle: 221
normals: 91 ,
"urves) UrT6
Optics: 1+0,203
Orthogonal trajectory: 223
Orthoptic: 3,73,138,139,11+9
Loria: 186
Osculinf lexion: 198, 195 , 1-00 , 202
Maclaurin: 11+3,160, 163,182, 205,
Ovals: 131,11+9,203
91,111,112,129,136,138,139,
11+0,11+9, 156, 157, 161, 163,161+,
168,169,173,176,162,183,187,
urve: 17O-I7I
Peaucelller cell: 10,28,52,131
Reflection- (see Caustics
Pedai Curves: 160-165)5,9,15,29,
65, 72,79, 85,136,138,ll+l+,ll+9,
167,179,182,203,207,209,211+,
Pedal Equations: l66-l69jl62,
177,213
Bhumb line: 118
Hlccati: 113
Rldolphl: 235
Eoberval: 65,66,11+8
Eoemer: 1,81,233
pin^fSe 2 *
Eooes: 85, 163,17!+, 216,235
Piriform: 201+
Roulettes: 175-185;13,29,6
Points, Singular: 192,199,200,
79, 110, 135, 136, ''07, 212,"
235,235 (see Trochoids)
Polars: l+l,te,l+3,l+l+,133
Polynomial Curves: 61+, 89, 19!+, 198
Polynomial Curves, Semi-: 61,87,
L^property: T
2kk INDEX
Singular points: 62,192,197, Sturm: 26
199,200,202 Suardi: 235
Sketching: 188-205:155
Slope: 191 Tangent Construction: 3,13,29,
Blot machine: 96 32, kl, hk ,'*6,6C, 73, 35, 119,139,
Sluze, Pearls of: 201+ H5, 150, 153, l68,21k,222
Snowflake Curve: 106 i'angeritB at origin: 191,192
Soap films: 13,183 Tautochrcne: 67,85
1 -06-216 Taylor: 75
Spirals, -:erquem: 160
Archimedean: 20, 156,16k, 1-59, :c,~.x s ; 9,20k
Cotes'; 72,169,215,216 137, 17k, 182, 20k, 212
Equiangular: 20,63,87,126, Trains: 2k
136,163,169,171,173,206, Trajectory, orthogonal: 223
207,208,209,211,216 Trammel of Archimedes: 3,77,108,
Euler: 136,215 120,23k
Fermat's: (see Spirals, Para- Transition curve: 56,215
Hyperbolic: (see Spirals, Trifolium: ( see Folium)
Eeciprocal) Trigonometric functions: 225-232
Parabolic: 169,212,213 Trisection: 33,36,58,205
Poinsot's- 20k Trisectrix: lk9, 163,203, 205
Eeciprocal: 182,210,211, Trochoids: 23?-236;120,122,158,
212,216,222 139, lk8, 176, 20k
Sinusoidal: 20,63, 139, IkO, Trophy: 20k
Ikk,l6l,l62,l63,l68,203, Tschirnhausen: 15,152,203,203,
213,21k 21k
Spiric Lines of Perseus: 20k Tulip: 186
21 9 Varignon: 211
Stubbs: 127 Vibration: 68,230,231,232
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